USF Libraries
USF Digital Collections

Kernel density estimation of reliability with applications to extreme value distribution

MISSING IMAGE

Material Information

Title:
Kernel density estimation of reliability with applications to extreme value distribution
Physical Description:
Book
Language:
English
Creator:
Miladinovic, Branko
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla
Publication Date:

Subjects

Subjects / Keywords:
Gumbel
Bayesian
Optimal bandwidth
Target time
Unbiased estimation
Dissertations, Academic -- Mathematics -- Doctoral -- USF   ( lcsh )
Genre:
non-fiction   ( marcgt )

Notes

Abstract:
ABSTRACT: In the present study, we investigate kernel density estimation (KDE) and its application to the Gumbel probability distribution. We introduce the basic concepts of reliability analysis and estimation in ordinary and Bayesian settings. The robustness of top three kernels used in KDE with respect to three different optimal bandwidths is presented. The parametric, Bayesian, and empirical Bayes estimates of the reliability, failure rate, and cumulative failure rate functions under the Gumbel failure model are derived and compared with the kernel density estimates. We also introduce the concept of target time subject to obtaining a specified reliability. A comparison of the Bayes estimates of the Gumbel reliability function under six different priors, including kernel density prior, is performed. A comparison of the maximum likelihood (ML) and Bayes estimates of the target time under desired reliability using the Jeffrey's non-informative prior and square error loss function is studied. In order to determine which of the two different loss functions provides a better estimate of the location parameter for the Gumbel probability distribution, we study the performance of four criteria, including the non-parametric kernel density criterion. Finally, we apply both KDE and the Gumbel probability distribution in modeling the annual extreme stream flow of the Hillsborough River, FL. We use the jackknife procedure to improve ML parameter estimates. We model quantile and return period functions both parametrically and using KDE, and show that KDE provides a better fit in the tails.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2008.
Bibliography:
Includes bibliographical references.
System Details:
Mode of access: World Wide Web.
System Details:
System requirements: World Wide Web browser and PDF reader.
Statement of Responsibility:
by Branko Miladinovic.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 144 pages.
General Note:
Includes vita.

Record Information

Source Institution:
University of South Florida Library
Holding Location:
University of South Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 002007059
oclc - 401321769
usfldc doi - E14-SFE0002760
usfldc handle - e14.2760
System ID:
SFS0027077:00001


This item is only available as the following downloads:


Full Text

PAGE 1

KernelDensityEstimationofReliabilityWithApplication stoExtremeValue Distribution by BrankoMiladinovic Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofMathematics&Statistics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:ChrisP.Tsokos,Ph.D. GangaramLadde,Ph.D. KandethodyRamachandran,Ph.D. MarcusMcWaters,Ph.D. DateofApproval: October16,2008 Keywords:Gumbel,Bayesian,optimalbandwidth,targettim e,unbiasedestimation c r Copyright2008,BrankoMiladinovic

PAGE 2

Dedication TomyfatherStanisavMiladinovic,whointroducedmetoMath ematicsandwhose lovemadeallofthispossible

PAGE 3

Acknowledgements Mydeepestappreciationgoesouttomymajorprofessorandme ntorProfessor ChrisTsokosforhishelpandencouragementduringmystudy. HisadviceandexpertiseintheeldofStatisticshaveshownmethemeritsofa cademicresearch.I wouldliketoexpressmygratitudetoProfessorsKandethody Ramachandran,GangaramLadde,andMarcusMcWatersfortheirserviceonmydiss ertationcommittee. IwouldliketothankProfessorEdwardMierzejowskiforchai ringmydissertation committee.Lastly,IwouldliketoacknowledgelateProfess orA.N.V.Raoforhisadviceduringthecourseofmystudy,andaboveallforbeingane xampleofawonderful humanbeing.

PAGE 4

TableofContents ListofFigures................................ivListofTables.................................viAbstract....................................ix1Introduction................................1 1.1BasicPropertiesoftheReliabilityFunction......... .1 1.2JusticationforBayesianAnalysis...............41.3TheGumbelFailureModel...................51.4TheNonparametricKernelDensityEstimateofReliabili ty.8 1.5ContentsofthePresentStudy.................11 2TheKernels:AnEvaluation.......................13 2.1Introduction...........................132.2KernelsandTheirProperties.................142.3EvaluationofKernelEectivenessinDensityEstimatio n..19 2.4NewRankingBasedonDierencesinOptimalBandwidthfor PDF, CDF,andReliabilityFunctions................22 VisualInspectionProceduretoDetermineOptimalBandwidt h forCDFandReliabilityFunctions..........24 2.5BandwidthRobustness.....................292.6Conclusion............................33 3Ordinary,Bayes,EmpiricalBayes,andKernelDensityReli abilityEstimates fortheGumbelFailureModel......................34 3.1Introduction...........................343.2TheGumbelFailureModel...................36 i

PAGE 5

3.3ReliabilityModeling......................373.4BayesEstimatorsofReliability................40 LindleyApproximationof ^h B ( t )...............43 3.5Non-parametricKernelDensityEstimatesofReliabilit y...47 3.6NumericalResults........................483.7Conclusion............................59 4SensitivityBehaviorofBayesianReliabilityfortheGumb elFailureModel forDierentPriors............................60 4.1Introduction...........................604.2ThePriors............................614.3MainResults..........................66 PropertiesofReliability...................70 4.4NumericalComparisonofPriors................714.5Conclusion............................73 5BayesianModelingofTargetTimefortheGumbelFailureMod el:Random LocationandScaleParameters.....................74 5.1Introduction...........................745.2TheGumbelModel.......................745.3ReliabilityModeling......................755.4BayesianApproachtotheGumbelModel...........76 Jerey'sNon-informativePrior...............76PosteriorDistribution.....................77BayesianEstimationof t forJerey'sPrior........78 TheLindleyApproximation.................79 5.5NumericalAnalysis.......................825.6Conclusion............................86 6TheChoiceoftheLossFunctionUnderBayesianParameterEs timation87 6.1Introduction...........................876.2LossFunctionSelectionCriteria................88 ii

PAGE 6

Criterion1:MinimaxCriterionUsingPosteriorRisks...88Criterion2:Makov'sCriterion................88Criterion3:GoodnessofFitCriterion...........89Criterion4:ProbabilityDensityCriterion.........89 6.3MainResults..........................90 Criterion1:MinimaxCriterionUsingPosteriorRisks...92Criterion2:Makov'sCriterion................93Criterion3:GoodnessofFitCriterion...........94Criterion4:ProbabilityDensityCriterion.........94 6.4CriteriaComparison......................956.5Conclusion............................100 7KernelDensityEstimationasanAlternativetotheGumbelD istributionin ModelingQuantilesandReturnPeriodsforFloodPrevention ....102 7.1Introduction...........................1027.2PreliminaryExplorationoftheExtremeStreamFlowData .104 7.3PeakStreamFlowQuantileandReturnPeriodModeling..1 06 Model1:TheMaximumLikelihoodModel.........109Models2and3:TimeDependentLocationParameter...112Model4:TheJackknifeModel................114Model5:ABayesianModel.................116Model6:TheNon-parametricKernelDensityModel...118 7.4ModelComparisonandRecommendation...........1217.5Conclusion............................122 8FutureResearch..............................127References...................................129Appendices..................................138AbouttheAuthor..............................EndPage iii

PAGE 7

ListofFigures 2.1EpanechnikovKernel........................152.2CosineKernel............................152.3BiweightKernel...........................162.4TriweightKernel..........................162.5GaussianKernel...........................172.6TriangleKernel...........................172.7UniformKernel...........................182.8NumericalStudyofKernelRanking................273.1GammaPriorfor =5, =0.5.................50 3.2GammaPriorfor =5, =1..................50 3.3GammaPriorfor =5, =4..................51 3.4OptimalReliabilityforh=0.4286.................523.5NumericalStudyofGumbelReliability..............5 3 4.1NumericalStudyofPriors.....................655.1NumericalStudyoftheGumbelFailureTime..........846.1GoodnessofFitCriterionImplementationChart....... ..95 6.2KernelDensityCriterionImplementationChart....... ..96 7.1AnnualMaximaStreamFlowfortheHillsboroughRiver194 0-2006105 7.2Ninety-vePercentCondenceBandFrequencyFactorPlo tforthe AnnualPeakStreamFlow.....................107 iv

PAGE 8

7.3StreamFlowQuantileFunctionUndertheMLEstimateswit h95% CondenceBands..........................111 7.4StreamFlowReturnPeriodFunctionUnderML.........11 2 7.5StreamFlowQuantileFunctionUnderJackknife........ .116 7.6StreamFlowReturnPeriodFunctionUndertheJackknifeM odel116 7.7StreamFlowQuantilefunctionUnderJackknifeandBayes Models118 7.8ReturnPeriodFunctionUnderJackknife,Bayes,andKern elDensity Models................................120 v

PAGE 9

ListofTables 1.1MostCommonlyUsedKernels...................92.1KernelsandTheirIneciencies..................212.2RateofDecreaseofAMISEForAllSevenKernelsasaFuncti onof SampleSize.............................21 2.3MeanIntegratedSquareError(MISE)fortheTopThreeBan dwidths forDataFromGumbel(n=15,30,50,100,200, =1)....28 2.4MeanIntegratedSquareError(MISE)fortheTopThreeBan dwidths forDataFromGumbel(n=15,30,50,100,200, =2)....29 2.5MeanIntegratedSquareError(MISE)fortheTopThreeBan dwidths forDataFromGumbel(n=15,30,50,100,200, =4)....30 2.6AMISERateofChangefor h ,n=20..............31 2.7AMISERateofChangefor h ,n=50..............31 2.8AMISERateofChangefor h ,n=100.............32 3.1Generated ValuesUndertheGammaPriorWith =5, =0.549 3.2Generated ValuesUndertheGammaPriorWith =5, =149 3.3Generated ValuesUndertheGammaPriorWith =5, =449 3.4MISEfortheReliabilityEstimates................553.5MISEfortheFailureRateFunctionEstimates.......... 56 3.6MISEfortheCumulativeFailureFunctionEstimates.... ..57 3.7MISEfortheTargetTime t c ...................58 4.1MISEUnderInverseGaussianPrior...............67 vi

PAGE 10

4.2MISEUnderInvertedGammaPrior...............684.3MISEUnderGammaPrior.....................684.4MISEUnderGeneralUniformPrior...............694.5MISEUnderDiusePrior.....................704.6AverageIntegratedMeanSquareErrorsfor =1........72 4.7AverageIntegratedMeanSquareErrorsfor =2.......72 4.8AverageIntegratedMeanSquareErrorsfor =4........72 5.1ComparisonBetweenMLandBayesianEstimatesofReliabi lityTime: N(25,1), =1,2,4, =0.01................84 5.2ComparisonBetweenMLandBayesianEstimatesofReliabi lityTime: N(25,2), =1,2,4, =0.01................85 5.3ComparisonBetweenMLandBayesianEstimatesofReliabi lityTime: N(25,3), =1,2,4, =0.01................85 6.1PriorParameterValues.......................976.2PercentageofSuccessoftheDierentCriteria........ ..97 6.3NumericalComparisonoftheDierentCriteriaUsedfort heChoiceof theLossFunction..........................100 7.1MLEstimatesoftheLocationandShapeParameterEstimat esand GoodnessofFit...........................111 7.2Log-likelihoodEstimatesforMLLinearandQuadraticTr endModels..................................113 7.3LocationandShapeParameterEstimatesUndertheJackkn ifeModel andGoodnessofFit........................115 7.4GoodnessofFitP-valuesfortheKernelDensityMethodfo rtheTop KernelandOptimalBandwidth..................120 7.5DierencesBetweentheEmpirical,andJackknife,Bayes ,andKernel DensityCDFEstimatesfortheTopEightTailValues......12 1 vii

PAGE 11

7.6MajorQuantilesfortheAnnualPeakStreamFlowUnderthe TopFour Models................................122 7.7HillsboroughRiverAnnualPeakStreamFlowNearZephyrh ills,FL.126 viii

PAGE 12

KernelDensityEstimationofReliabilitywithApplications to ExtremeValueDistribution BrankoMiladinovic Abstract Inthepresentstudy,weinvestigatekerneldensityestimat ion(KDE)anditsapplicationtotheGumbelprobabilitydistribution.Weintroduc ethebasicconceptsof reliabilityanalysisandestimationinordinaryandBayesi ansettings.Therobustness oftopthreekernelsusedinKDEwithrespecttothreedieren toptimalbandwidths ispresented.Theparametric,Bayesian,andempiricalBaye sestimatesofthereliability,failurerate,andcumulativefailureratefunctio nsundertheGumbelfailure modelarederivedandcomparedwiththekerneldensityestim ates.Wealsointroduce theconceptoftargettimesubjecttoobtainingaspeciedre liability.Acomparison oftheBayesestimatesoftheGumbelreliabilityfunctionun dersixdierentpriors, includingkerneldensityprior,isperformed.Acomparison ofthemaximumlikelihood(ML)andBayesestimatesofthetargettimeunderdesire dreliabilityusingthe Jerey'snon-informativepriorandsquareerrorlossfunct ionisstudied.Inorder todeterminewhichofthetwodierentlossfunctionsprovid esabetterestimateof thelocationparameterfortheGumbelprobabilitydistribu tion,westudytheperformanceoffourcriteria,includingthenon-parametrickerne ldensitycriterion.Finally, weapplybothKDEandtheGumbelprobabilitydistributionin modelingtheannual extremestreamrowoftheHillsboroughRiver,FL.Weusethej ackknifeprocedure toimproveMLparameterestimates.Wemodelquantileandret urnperiodfunctions bothparametricallyandusingKDE,andshowthatKDEprovide sabettertinthe tails. ix

PAGE 13

1Introduction Inthepresentstudy,wewillinvestigatenon-parametricke rneldensityestimation anditsapplicationtoanextremevaluedistribution,namel ytheGumbelprobability distribution.Theprimaryobjectiveistointroducenon-pa rametrickerneldensity methodology,inordertoestimatetheprobabilitydensityf unctionofgivendata whenwecannotidentifyaclassicalprobabilitydensityfun ction.Thesubjectnonparametrickerneldensityisafunctionoftwoimportantqua ntities,namelythekernel functionandtheoptimalbandwidth.Wewillevaluatesevenm ostcommonlyused kernelsandrankthetopthreewithrespecttotheireective ness,whichwillbemeasuredbythesizeofthemeansquareerror(MSE),inconjuctio nwithaselected optimalbandwidth.Thismethodologywillbeappliedtoreli abilityanalysiswhenwe cannotidentifythefailureprobabilitydensityfunctiono fagivensystem.Wewill showthatthisnon-parametricstatisticalprocedureisqui teeectivewhencompared toparametricanalysis.Weproceedtointroducesomeprelim inarydenitionsand proceduresthatwillbeusedinthepresentstudy. 1.1BasicPropertiesoftheReliabilityFunction Let T 1 T 2 ,..., T n bearandomsampleofsizentakenfromapopulationofinteres t, where T 1 T 2 ,..., T n areindependentandidenticallydistributed.Acompletech aracterizationoftherandomvariablebeingobservedisgivenon lyifwecanspecifyexactly itsprobabilitydistributionfunction.Afunctionwithval uesf(t),denedovertheset ofallrealnumbers,iscalledaprobabilitydensityfunctio n(PDF)ofthecontinuous 1

PAGE 14

randomvariableTifandonlyif P ( a T b )= Z b a f ( t ) dt (1.1.1) foranyrealconstantsaandbwitha b.Afunctioncanserveasaprobabilitydensity ofacontinuousrandomvariableTifitsvalues,f(t),satisf ytheconditions f ( t ) 0 ; 1 t + 1 (1.1.2) and Z + 1 1 f ( t ) dt =1(1.1.3) ThefunctionF(t)=P(T t)iscalledthecumulativedistributionfunction(CDF) oftherandomvariableT.IfTdenotesthetimetofailureofap articularsystemfrom time0totimet,thenthereliabilityfunctionR(t)isthepro babilitythatasystem willbeoperableuptotimetandisdenedas R ( t )= P ( T>t )=1 F ( t )= Z 1 t f ( x ) dx (1.1.4) wheref(t)istheprobabilitydensityfunctionoftherandom variableT.Thevalues R(t)ofthereliabilityfunctionofarandomvariableTsatis fytheconditions: R (0)=1and R (+ 1 )=0(1.1.5) and if a
PAGE 15

weobtaintheexpressionwhichforagivensystemrepresents the"targettime"to failure t withatleast(1)100%condence.Alsorelatedtoreliabilityarethe failurerateandcumulativefailureratefunctions.Thefai lureratefunction,h(t),is theprobabilitythatagivensystemwillfailforthersttim eaftertimet,giventhat ithasoperateduptotimet.UnderthereliabilityfunctionR (t),thedenitionsof thefailureratefunctionh(t)andcumulativefailureratef unction,H(t),aregiven respectivelyby h ( t )= @R ( t ) @t R ( t ) (1.1.8) and H ( t )= ln ( R ( t ))(1.1.9) Whenengagedinreliabilityanalysis,given T 1 T 2 ,..., T n failuretimes,werstproceed throughgoodness-of-ttestsandpossiblylteringmethod stoidentifyparametersof aclassicalwelldenedprobabilitydensityfunctionthatp robabilisticallycharacterizesthebehavioroffailuretimes.Onlyifwearenotabletoi dentifysuchadensity function,weproceednon-parametrically.Theproblemsofr eliabilityforaspecied probabilitydistributionhavereceivedagreatdealofatte ntioninthepastyears.It hasbeenofparticularinteresttoobtainaminimumvariance unbiased(MVU)estimatorofreliabilityduetoconsiderablecumulativeeects ofbiasincomplexsystems. Forinstance,Pugh(1964)obtainedMVUestimatesofreliabi lityfortheexponentialfailuremodel.Tate(1959)andBasu(1964)derivedMVUe stimatorsofthe scaleparameterandthereliabilityfunctionfortheexpone ntial,Weibull,andGamma probabilitydistributionfunctions.Glasser(1962)deriv edMVUestimatorsforthe Poissonprobabilitydistribution.TheMVUestimatorswere derivedbyusingfunctionsofcompleteandsucientstatistics,asshallweinour study.Ourmainfocus willbeontheGumbelfailuremodelinordinaryandBayesians ettings,soweproceed todiscussBayesianmethodologynext. 3

PAGE 16

1.2JusticationforBayesianAnalysis Bayesianinferenceprocedureshavegainedwidespreadpopu larityinrecentyears; however,justicationisseldomgivenwhensuchprocedures areusedbyscientists andengineers.Bayesiananalysisrestsontheideathatifas cientistperformsan experiment,newstatisticalinferencescanbebuiltuponea rlierunderstandingofa phenomenonunderstudy.Italsoprovidesamethodologytofo rmallycombinethat earlierunderstandingwithcurrentlymeasureddata,sotha titupdatesthedegree ofbelief(subjectiveprobability)oftheexperimenter.Th eearlierunderstandingis calledthe"priorbelief,"whichistheunderstandingheldp riortoobservingthecurrentsetofdata,availablefromtheexperimenterorotherso urces.Thenewbelief, whichresultsfromupdatingthepriorinformation,iscalle dthe"posteriorbelief." Thisisthenewupdatedunderstandingheldafterhavingobse rvedthecurrentdata, examinedinlightofhowwelltheyconformwithpreconceived notions.Ifwefeelcondentthatthepriorinformationderivedfromearlierexper imentsmayimproveour reliabilityestimateswhenperformingreliabilityanalys is,thenBayesianmethodology ismoreappropriate.Ifweassumethatparameterswithinfai luremodelsbehaveas randomvariablesindividuallyorjointlyfollowingcertai ndistributions,wemayuse Bayesiananalysis,whichexploitsasuitablepriorinforma tionandthechoiceofaloss functioninassociationwithBayes'Theorem.Thetheoremsh owsusthattond subjectiveprobabilityforsomeeventorunknownquantity, weneedtomultiplyour priorbeliefsabouttheeventbyanappropriatesummaryofth eobservationaldata. Thisassumptionmaywellbejustied.Throughexperimentat ionwemaynoticethat duetothecomplexityofelectronicandstructuralsystems, undetectedcomponent interactionsresultinginunpredictableructuationsofth eparametersarepresent.To areliabilityengineerthisapproachwouldseemtobeappeal ingbecauseitprovidesfor theformulationofadistributionalformforanunknownpara meterbasedontheprior convictionsoravailableinformation,especiallyasrelia bilitypredictiontechniquesare basedonpooledandorganizedexperienceofcountlessindiv idualsandorganization. AnothertypeofBayesiananalysis,introducedbyRobbins(1 980),involvesestimating 4

PAGE 17

aparameterofadatadistributionwithoutknowingorassess ingtheparametersofthe subjectivepriorprobabilitydistribution.Thisanalysis ,calledempiricalBayes,estimatesthepriordistributionofaparameterdirectlyfromth edata.Inourstudywewill engageinbothclassicalandempiricalBayesanalysis.Crel in(1972),Drake(1966), andEvans(1969)giveexcellentphilosophicaljusticatio nsfortheuseofBayesian methodologiesinreliabilityanalysis.Tsokos(1972)gave anexcellentreviewofthe Bayesianapproachtoreliabilityandclearlyexpressedthe usefulnessofMonteCarlo simulation.CavanosandTsokos(1970)introducedtheconce ptofempiricalBayes approachtoreliabilityfortheWeibullfailuremodel.Cava nosandTsokos(1971) derivedclassicalandempiricalBayesestimatesforthepar ameterandreliabilityfor theGammafailuremodel,alongwithMonteCarlonumericalst udytoillustratethe sensitivityoftheestimates.Becausethephilosophybehin dempiricalBayesestimationrestsontheassumptionthattheexistenceofapriordis tributionisknown,but itsformisunknown,ourstudywillinvolvebothclassicalBa yesandempiricalBayes methodologies,aswellasMonteCarlosimulation,toillust ratetheusefulnessofour estimates.InourstudythemainfocuswillbeontheGumbelfa iluremodel,which wediscussnext. 1.3TheGumbelFailureModel Fromprobabilitytheory,forthelargestnumberofnindepen dentidenticallydistributedrandomvariables Y 1 ;Y 2 ;:::;Y n ,i.e., X := max ( Y 1 ;Y 2 ;:::;Y n ) theprobabilitydistributionfunctionisgivenby M n ( x )=[ F ( x )] n whereF(x)=P( Y i x)isthecommonprobabilitydistributionfunctionofeach of Y i .F(x)iscommonlyreferredtoasaparentprobabilitydistri bution.Ifnis 5

PAGE 18

notconstantbutrathercanberegardedasarealizationofar andomvariablewith Poissonprobabilitydistributionwithmean ,thenthedistributionofXbecomes (e.g.TodorovicandZelenhasic,1970;Rossietal.,1984), M 0 ( x )= exp f (1 F ( x )) g Sinceln[ F ( x )] n n (1 F ( x )),itfollowsthatforlargenorlargeF(x), M n ( x ) M 0 n (x).Gumbel(1958),followingthepioneeringworksbyFrche t(1927),Fisherand Tippet(1928)andGnedenco(1941),developedacomprehensi vetheoryofextreme valuedistributions.Thatis,asntendstoinnity, M n ( x )convergestooneofthree possibleasymptoticdistributions,dependingonthemathe maticalformofF(x).Obviously,thesamelimitingdistributionsmayalsoresultfr om M (x)as tendsto innity.Allthreeasymptoticdistributionscanbedescrib edbyasinglemathematical expressionintroducedbyJenkinson(1955,1969)knownasth eGeneralizedExtreme Value(GEV)probabilitydistributionfunction.Thisexpre ssionisgivenby M ( x )=exp f (1+ k ( x ) ) 1 =k g ;kx k (1.3.1) where > 0andkarelocation,scaleandshapeparameters,respective ly.When k=0,thetypeIprobabilitydistributionofmaxima(EV1orGu mbeldistribution)is obtainedasaspecialcaseoftheGEVdistribution.Usingsim plecalculusitisfound thatinthiscase,thecumulativedistributionfunctiontak estheform M ( x )= exp ( exp ( x ))(1.3.2) whichisunboundedfrombothleftandright.Therefore,fora sampleofnrandomiid failuretimes, T 1 T 2 ,..., T n ,thereliabilityfunctionundertheGumbelfailuremodel isgivenby R ( t )=1 exp ( exp ( t ))(1.3.3) t> 0 ; 1 << 1 ;> 0 6

PAGE 19

Incidentally,whenk > 0,M(x)representstheextremevalueprobabilitydistribut ion ofmaximaoftypeII(EV2).Inthiscasethevariableisbounde dfromtheleftand unboundedfromtheright( /k=x < 1 ).Aspecialcaseisobtainedwhenthe leftboundbecomeszero( =k ).Thisspecialtwo-parameterdistributionhasbeen knownastheFrchetdistributionandhasthesimpliedform M ( x )= exp ( ( x ) 1 =k ) ;x 0 with becomingascaleparameter.SincetheGEVprobabilitydistr ibutioninvolves threeparameters,italwaysprovidesbetterestimatesthan theGumbelprobability distribution;however,practically,itisalsomoredicul ttoworkwithduetotheanalyticconstraintsinvolvedinthreeversustwoparameteres timation,astwoparameters aremoreaccuratelyestimatedthanthree.Duetoitssimplic ityandgenerality,the Gumbelprobabilityhasbeenintroducedasafailuremodelfo rreliabilitystudies.A recentbookbyKotzandNadarajah(2000)listsover50applic ationrangingfrom acceleratedlifetestingthroughtoearthquakes,roods,ho rseracing,rainfall,queues insupermarkets,seacurrents,windspeeds,andtrackracer ecords.Inparticular,the Gumbelprobabilitydistributionhasbeenusedtocharacter izerealworldproblemsfor severalreasons.Mosttypesofparentprobabilitydistribu tionsfunctionsthatareused inreliability,suchasexponential,Gamma,Weibull,norma l,lognormal(Kottegoda andRosso(1997))belongtothedomainofattractionoftheGu mbeldistribution.In contrast,thedomainofattractionoftheGEVdistributioni ncludeslesscommonly metparentprobabilitydistributionslikePareto,Cauchy, andlog-gamma.Developed inthe1950's,goodness-of-tprobabilityplotsarethemos tcommontoolsusedby practitionersandengineerstochooseanappropriatedistr ibutionfunction.EV1oers alinearGumbelprobabilityplot,whichisestimatedinterm sofplottingpositions, i.e.sampleestimatesofprobabilityofnon-failure.Incon trast,alinearprobability plotforthethree-parameterGEVisnotpossibletoconstruc t.Thismayberegarded asaprimaryreasonofchoosingEV1againstthethree-parame terGEVinpractice, assumingthattwoparametersproduceresultsalmostasgood asthree.Forthefor7

PAGE 20

mercase,meanandstandarddeviation(orsecondL-moment)s uce,whereasinthe lattercasetheskewnessisalsorequiredanditsestimation isextremelyuncertainfor typicalsmall-sizereliabilitysamples.However,EV1haso nedisadvantage,whichis veryimportantfromtheengineeringpointofview:forsmall probabilitiesoffailure orexceedenceityieldsthesmallestpossiblequantilesinc omparisontothoseofthe three-parameterGEVforany(positive)valueoftheshapepa rameterk.Thismeans thatEV1resultsinthehighestpossibleriskforengineerin gstructures(Farquharson etal.(1992),Turcotte(1994),TurcotteandMalamud(2003) ).Wewillestablishin chapter7thatkerneldensityestimationprovidesanaltern ateandrelativelyeasyway toremedythispotentialdrawbackoftheGumbelprobability distribution.Asstated earlier,ifwearenotabletoidentifytheunderlyingprobab ilitydensityfunctionfor agivendatasetparametrically,wemaydosonon-parametric ally.Weproceedto introducenon-parametrickerneldensityestimationasapo werfulalternative. 1.4TheNonparametricKernelDensityEstimateofReliabili ty Themainproblemwiththeparametricapproachisthatexisti ngclassicalprobability distributionfamiliesarelimitedinthefaceofamultitude ofdatastructures.Awrong assumptionconcerningtheunderlyingdistributionmodelf orthedatamayleadto misleadinginterpretations.Insituationssuchasthese,n on-parametricmethodsmay bemoresuitable.Thenonparametricmethodsimposeonlymil dassumptions,such assmoothness,ontheunderlyingprobabilitydistribution andsoavoidtheriskof specifyingthewrongmodelforthedata.Thereareseveraldi erentmethodsin nonparametricprobabilitydensityestimation,suchasker nelandorthogonalseries estimates(Silverman(1986)),maximumpenalizedlikeliho odestimates(Tapiaand Thompson(1974)),smoothingsplines(Gu(1993)),wavelete stimates(Donohoet.al. (1996)),andother.Amongthese,kernelestimatesaremostw idelyusedandeasiest toimplement.Ourstudywillconcentrateonestimatingther eliabilityfunctionusing thekernelapproach.Kerneldensityestimationhasbeenapp liedtosuchdiverseelds asEconomics,Ecology,Neurocomputing,WildlifeManageme nt,NeuralNetworksand 8

PAGE 21

Populationresearch.Forthebenetofthereader,overfty majorandmostrecent publicationsintheeldofkerneldensityestimationareli stedchronologicallyin AppendixI.Let T 1 T 2 ,..., T n bei.i.d.randomvariableshavingacommonPDFf(t).Thekern el densityestimate(KDE)oftheprobabilitydensityfunction f(t)isgivenby: ^ f n ( t )= 1 nh n X i =1 K ( t T i h )(1.4.1) whereK(u)isthekernelfunctionandhisthebandwidth.Thek ernelfunctionKis usuallyrequiredtobeasymmetricprobabilitydensityfunc tion,whichmeansthatK satisesthefollowingconditions Z + 1 1 K ( u ) du =1 ; Z + 1 1 uK ( u ) du =0 and Z + 1 1 u 2 K ( u ) du> 0 : (1.4.2) PropertiesofkernelfunctionKdeterminethepropertiesof theresultingkernelestimates,suchascontinuityanddierentiability.Tradition ally,sevenkernelfunctions havebeenusedinnon-parametrickerneldensityestimation andaregivenintable 1.1.Inourstudy,wewillevaluatetheeectivenessoftheke rnels,providearanking Kernel Form Epanechnikov 3 4 (1 u 2 ) I ( j u j 1) Cosine 4 cos ( u 2 ) I ( j u j 1) Biweight 15 16 (1 u 2 ) 2 I ( j u j 1) Triweight 35 32 (1 u 2 ) 3 I ( j u j 1) Gaussian 1 p 2 e 0 : 5 u 2 Triangle (1 j u j ) I ( j u j 1) Uniform 0 : 5 I ( j u j 1) Table1.1:MostCommonlyUsedKernels dierentfromtheonecommonlyaccepted,andapplyourresul tsinreliabilitymod9

PAGE 22

eling.For T 1 T 2 ,..., T n i.i.d.randomvariableshavingacommonprobabilitydensit y functionf(t),thekerneldensityestimateofreliabilityi sdenedby ^ R n ( t )=1 1 nh n X i =1 Z t 1 K ( y T i h ) dy (1.4.3) Bysolvingforthequantile t forwhich 1 nh n X i =1 Z t 1 K ( y T i h ) dy = (1.4.4) wegetthenonparametricestimateof"targettime"tospeci edreliability(1)100%, where isverysmall.Likewise,thenonparametricestimateofthef ailurerateand cumulativefailureratefunctionsaregivenby: ^ h n ( t )= @ ^ R n ( t ) @t ^ R n ( t ) (1.4.5) and ^ H n ( t )= ln ( ^ R n ( t ))(1.4.6) SincethekerneldensityestimatewasintroducedbyRosenbl att(1956),manyapproachestobandwidthselectionhavebeenproposed(seeBea nandTsokos(1980), Marron(1989),Silverman(1986),Simono(1996),WandandJ ones(1995)).Broadly speaking,databasedoptimalbandwidthselectionproposal scanbedividedinto"rst generation"(seeMarron(1989)and"secondgeneration"(se eJoneset.all(1995)). Most"rstgeneration"methodsweredevelopedpriorto1990 andinclude"Rules ofThumb,""LeastSquaresCross-Validation","BiasedCros s-Validation.""Second generation"methodssuchas"solveandpluginmethod,"and" smoothedbootstrap method"havebeenproposedfortheirsuperiorperformanceo verthe"rstgeneration" methods,whichwasshownbyJoneset.al(1995)byasymptotic analysis,simulation, andrealdatastudy.However,alloftheseapproacheshavece nteredonthekernel densityestimationoftheprobabilitydensityfunctionand notenoughonkernelden10

PAGE 23

sityestimatesofthecumulativedensityorreliabilityfun ction.Someworkhasbeen doneregardingconsistencyofkernelCDFestimates(seeNad araya(1964),Winter (1973),andYamato(1973))andselectionofbandwidthfromt hetheoreticalpoint ofview(Sarda(1990)),butverylittleregardingpractical implementationbasedon thedierencebetweenoptimalbandwidthforPDFandCDF(see LiuandTsokos (2002)).Inthisstudy,wewillusesomeoftheresultsofLiua ndTsokos(2002)to studythechoiceofkernelwithrespecttothechoiceofoptim albandwidthandsample size.Wewillproposeanewrankingofkernelsbasedontheire cienciesinchapter 2andapplyourresultstothestudyoftheGumbelmodelinsubs equentchapters. 1.5ContentsofthePresentStudy Inthissectionwelistthecontributionsofourstudyandasu mmaryofourresults. Inchapter2wediscusstheselectionofoptimalbandwidthfo rreliabilityunderkerneldensityestimation.Ourextensivenumericalsimulatio nshowsadierentranking ofkerneleciencythancommonlyaccepted(Silverman(1986 )).Weshowthatthe topthreekernelsarerobustwithrespecttotheoptimalband widthandsamplesize. Inchapter3wemodifytheclassicGumbelordoubleexponenti aldistributionand deriveestimatesofreliabilityparametricallyinordinar y,BayesandempiricalBayes settings(MiladinovicandTsokos(2008)).Usingnumerical simulation,wecompare theparametricestimatesofGumbelreliabilitywiththeirn on-parametrickerneldensitycounterparts,whicharederivedusingthemethodology fromchapter2.Weshow thatkerneldensityestimatesperformaswellastheparamet ricones.Chapter4 presentsastudyofrobustnessofGumbelreliabilityunderd ierentpriors,including ourownkerneldensityprior.Wederiveandnumericallycomp aretheGumbelreliabilityestimatesunderdierentpriorsandshowthattheyar erobust.Inchapter5we deriveBayesianestimatesofthetargettimesubjecttoaspe ciedreliabilityforthe Gumbelfailuremodelandshowthatitprovidescloserestima testhanthemethodof maximumlikelihood(MiladinovicandTsokos(2008)).Sever alcriteriaforthechoice ofthelossfunctioninBayesiananalysisarecomparedwitho urownkerneldensity 11

PAGE 24

criterioninchapter6.Weshowthatourcriterionismostcon sistentincomparing Bayesianestimates.Inchapter7,weapplyboththeGumbeldi stributionandKDE tothemodelingofrooddataandshowthattheKDEprovidesabe ttertinthetails, whichmakesitmoreappropriateinmodelingtheveryextreme occurrencesthanthe Gumbelmodel.Finally,inchapter8wepresentplansforfutu reresearch. 12

PAGE 25

2TheKernels:AnEvaluation 2.1Introduction Kerneldensityestimation(KDE)playsanimportantroleint heprobabilisticcharacterizationofphenomenawhenweareunabletoidentifyawe ll-denedprobability densityfunction(PDF)intheparametricsense.Somekeyref erencesareChen(1999), DiNardoandTobias(2001),Goutis(1997),Padget(1988),Po wellandSeaman(1996), QiaoandTsokos(1992).Instudyingreliabilityofagivensy stem,anengineermay notbeabletoidentifyawelldenedprobabilitydistributi onfunctionandrelyon kerneldensityestimationasanalternative.Findingthebe stkerneldensityestimate ofreliabilitywouldinvolvethemostimportantaspectofke rneldensityestimation, whicharethechoiceoftheappropriatekernelandthecorres pondingoptimalbandwidth.Silverman(1986)evaluatedandrankedsevenmajorke rnelsbasedonaspecic (optimal)bandwidth.Theobjectiveofthepresentstudyist oevaluatethesubject kernelsbyusingamorerexiblebandwidthandalsobyvarying thesamplesizen. WewillshowthatourrankingisbetterthanSilverman's.Mor especically,wewill accomplishthefollowing: (i)Insection2.2,discussthepropertiesofthesevenmajor kernels. (ii)Insection2.3,discusshowtomeasuretheeectiveness ofkernelestimatesusing theasymptoticmeanintegratedsquareerror(AMISE).Wewil lusetherateof decreaseofAMISEtorankthekernelsasafunctionofthesamp lesizenand discussSilverman'srankingofthekernelfunctionsandthe choiceofbandwidth. (iii)Insection2.4,presentanewrankingofthetopthreeke rnelsbasedonanew, 13

PAGE 26

morerexiblechoiceofbandwidthandourextensivenumerica lstudyofthe Gumbelfailuremodel.Wewillalsosuggesttheusageofdier entkernelsbased onthesamplesizen. (iv)Insection2.5,studyhowrobustthechangeofbandwidth iswithrespecttothe choiceofaspecickernel.Weprovideanextensivenumerica lstudytoevaluate thechangeinAMISEforeachkernel,astheoptimalbandwidth isvariedinequal intervalsfromtheoptimalvalue.Weconcludethatthetopth reekernelsare robustwithrespecttosignicantpercentchangefromtheop timalbandwidth. (v)InSection2.6,presentconcludingremarks. 2.2KernelsandTheirProperties Recallthatif X 1 ,... X n arei.i.d.randomvariableshavingacommonPDFf(x),the kernelestimateoff(x)isdenedby ^ f n ( x )= 1 nh n X i =1 K ( x X i h )(2.2.1) wherehisthebandwidthandK(u)isthekernelfunction.Thek ernelestimateof thecumulativedistributionfunction(CDF)F(x)andreliab ilityfunctionR(x)are respectivelygivenby ^ F n ( x )= 1 nh n X i =1 Z x 1 K ( y X i h ) dy (2.2.2) and ^ R n ( x )=1 ^ F n ( x )(2.2.3) FortherestofthestudyweshallassumethatK(u)isasymmetr icfunction.There aresevencommonlyusedkernelsfunctions.Theiranalytice xpressionaregivenin equations(2.2.4)-(2.2.10)withthecorrespondinggraphs giveninFigures1-7. 14

PAGE 27

EpanechnikovKernel 3 4 (1 u 2 ) I ( j u j 1)(2.2.4) 1 0.80.60.40.2 0 1 0.5 0 -0.5 -1 Epanechnikov Kernel Figure2.1:EpanechnikovKernel CosineKernel 4 cos ( u 2 ) I ( j u j 1)(2.2.5) 1 0.80.60.40.2 0 1 0.5 0 -0.5 -1 Cosine Kernel Figure2.2:CosineKernel 15

PAGE 28

BiweightKernel 15 16 (1 u 2 ) 2 I ( j u j 1)(2.2.6) 1 0.80.60.40.2 0 1 0.5 0 -0.5 -1 Biweight Kernel Figure2.3:BiweightKernel TriweightKernel 35 32 (1 u 2 ) 3 I ( j u j 1)(2.2.7) 1 0.5 0 -0.5 -1 1.41.2 1 0.80.60.40.2 0 Triweight Kernel Figure2.4:TriweightKernel 16

PAGE 29

GaussianKernel 1 p 2 e 0 : 5 u 2 (2.2.8) 0.50.40.30.20.1 0 4 2 0 -2 -4 Gaussian Kernel Figure2.5:GaussianKernel TriangleKernel (1 j u j ) I ( j u j 1)(2.2.9) 1 0.80.60.40.2 0 1 0.5 0 -0.5 -1 Triangle Kernel Figure2.6:TriangleKernel 17

PAGE 30

UniformKernel 0 : 5 I ( j u j 1)(2.2.10) ThesubjectkernelK(u)mustsatisfythefollowingproperti es: 1 0.80.60.40.2 0 1 0.5 0 -0.5 -1 Uniform Kernel Figure2.7:UniformKernel Z + 1 1 K ( u ) du =1 ; Z + 1 1 uK ( u ) du =0 andk 2 = Z + 1 1 u 2 K ( u ) du> 0 : (2.2.11) PropertiesofthekernelfunctionK(u)partiallydetermine thepropertiesofthekernel densityestimates,suchasdierentiabilityandcontinuit y.Forexample,ifK(u)isa properdensityfunction,thatisifitisnon-negativeandit integratestoone,then thekerneldensityestimateisalsoaproperdensityfunctio n.IfK(u)isntimes dierentiable,sois ^ f n .Epanechnikovsuggestedtherstkerneltobeusedinthe contextofdensityestimationin1956.Thegraphisgivenin gure2.1.Before Epanechnikovandinadierentcontext,HodgesandLehmann( 1956)showedthatthe kerneloptimizestheexpressionusedinndingtheoptimalb andwidth,thusmaking itthemostecientkernel.Historically,aftertheintrodu ctionoftheEpanechnikov kernelindensityestimation,thenon-smoothUniformandTr ianglekernelswerealso introducedasalternatives.Thesearchforsmoothandlessi necientkernelsproduced theremainingfour,theCosinekernelbeingaddedlast.Them ostpopularkernelin practicehasbeentheGaussiankernelduetoitsanalytictra ctability. 18

PAGE 31

2.3EvaluationofKernelEectivenessinDensityEstimatio n Silverman(1986)evaluatedsubjectkernelsK(u)bycompari ngthemwiththeEpanechnikovkernelandtheoptimalbandwidthgivenby h optimal =[ C ( K ) k 2 2 R ( f 00 ) ] 1 5 n 1 5 (2.3.1) where C ( K )= R K ( u ) 2 du k 2 = R + 1 1 u 2 K ( u ) du> 0,andf(x)isthedensityfunction tobeestimated.Hederivedtheformulafortheoptimalbandw idthbyutilizingthe measureofdiscrepancyofthedensityestimator ^ f ( x )atasinglepointcalledthe meansquareerror(MSE),whichisdenedby: MSE ( ^ f ( x ))= E ( ^ f ( x ) f ( x )) 2 =( E ^ f ( x ) f ( x )) 2 + var ^ f ( x ) : (2.3.2) where( E ^ f ( x ) f ( x ))isalsoreferredtoasthebiasof ^ f ( x ).Ifh 0andnh !1 andtheunderlyingdensity f 00 issucientlysmoothandabsolutelycontinuousand f 000 squareintegrable,thenitcanbeshownthat bias ( ^ f ( x ))= h 2 k 2 f 00 ( x ) 2 + o ( h 2 )(2.3.3) var ( ^ f ( x ))= f ( x ) C ( K ) nh + o ( 1 nh )(2.3.4) whereC(K)= R K ( u ) 2 du and k 2 = R + 1 1 u 2 K ( u ) du> 0.Fromexpressions(2.3.3) and(2.3.4)wecaninferthatifthebandwidthdecreases,the biasofthekernelestimatealsodecreasesbutthevarianceincreases,resultingi naroughandunacceptable estimateofthekerneldensity.Conversely,ifthebandwidt hincreases,thevariance ofthekernelestimatedecreasesbutthebiasincreases.Thi smeansthatthereissignicantsmoothingandtheunderlyingcharacteristicsofth eprobabilitydensityare smoothedout.Combiningexpressions(2.3.3)and(2.3.4)an dintegratingoverthe entirereallinegivesusanestimateoftheglobalaccuracyo f ^ f ( x ),theasymptotic 19

PAGE 32

meanintegratedsquareerror(AMISE): AMISE ( ^ f ( x ))= h 4 k 2 2 R ( f 00 ) 4 + C ( K ) nh (2.3.5) Thus,wecanconcludethatAMISEdependsonfourquantities: thebandwidthh,the samplesizen,kernelfunctionKandthetargetdensityf(x). Thetargetfunctionand thesamplesizeareoutofourcontrol,howeverwecanminimiz eAMISEbychoosing theappropriatekernelandthebandwidth.Ifwexthekernel functionK(u)and minimizeAMISEwithrespecttothebandwidthweobtain: h optimal =[ C ( K ) k 2 2 R ( f 00 ) ] 1 5 n 1 5 (2.3.6) and AMISE optimal = 5 4 ( p k 2 C ( K )) 4 5 C ( f 00 ) 1 5 n 4 5 (2.3.7) Tocalculatetheoptimalkernelfunction,weminimize AMISE optimal withrespect toK.Thismeansminimizing p k 2 C ( K )withoutknowingf(x).Theoptimalkernel functionwasderivedbyEpanecnikov(1969)andisgivenby K ( u )= 3 4 (1 u 2 ) I ( j u j 1) Thevalueof p k 2 C ( K )fortheEpanechnikovkernelis 3 5 p 5 ,sothattheratio p 125 k 2 C ( K ) 3 providesameasureofineciencyforotherkernels.Silverm an'srankingofallkernels accordingtotheirinecienceswhichisbasedontheoptimal bandwidthisgivenin table2.1.Wehavestudiedthe AMISE optimal asafunctionofnforallsevenkernels andrankedthemaccordingtothevalueofAMISEforeachkerne l.Ourresults indicatethatAMISEconvergestozeroapproximatelyunifor mlyforallsevenkernels. ThesizeofAMISEforeachkernelisgivenintable2.2.Weobta inthesameranking 20

PAGE 33

Kernel Form k 2 Ineciency Epanechnikov 3 4 (1 u 2 ) I ( j u j 1) 0.2 1.0000 Cosine 4 cos ( u 2 ) I ( j u j 1) 0.1894 1.0005 Biweight 15 16 (1 u 2 ) 2 I ( j u j 1) 0.1429 1.0061 Triweight 35 32 (1 u 2 ) 3 I ( j u j 1) 0.1111 1.0135 Gaussian 1 p 2 e 0 : 5 u 2 1.0000 1.0513 Triangle (1 j u j ) I ( j u j 1) 0.1667 1.0143 Uniform 0 : 5 I ( j u j 1) 0.3333 1.0758 Table2.1:KernelsandTheirIneciencies ifweorderthekernelsaccordingtotheirinecienciesaswe dowhenweorderthem accordingtotherateofdecreaseofAMISEasafunctionofsam plesizen.Silverman's Kernel n=50 n=100 n=150 n=200 n=500 Epanechnikov 0.0191 0.0109 0.0079 0.0063 0.0030 Cosine 0.0191 0.0110 0.0079 0.0063 0.0030 Biweight 0.0192 0.0110 0.0080 0.0063 0.0030 Triweight 0.0193 0.0111 0.0081 0.0064 0.0031 Gaussian 0.0199 0.0114 0.0082 0.0066 0.0031 Triangle 0.0193 0.0111 0.0080 0.0064 0.0030 Uniform 0.0202 0.0116 0.0084 0.0067 0.0032 Table2.2:RateofDecreaseofAMISEForAllSevenKernelsasa FunctionofSampleSize rankingissubjecttoseveralobjections.First,howdoesth esamplesizeaecteach kernel'sperformance?Second,howdothesekernelsrankifw eneedtoestimatethe cumulativedensityandreliabilityfunctions.Consistent withtheobjectiveofthis chapterwewillproceedtoevaluatethesekernelswithanewb andwidthdevelopedby LiuandTsokos(2002),andwithrespecttosmall,mediumandl argesamplesizen. 21

PAGE 34

2.4NewRankingBasedonDierencesinOptimalBandwidthfor PDF, CDF,andReliabilityFunctions InthissectionweturnourattentiontotheKDEofthecumulat ivedistribution functionF(x).ThepropertiesoftheKDEofF(x)willhelpuss tudytheKDEofthe reliabilityfunctionR(x),thefailurefunctionh(x)andth ecumulativefailurefunction H(x)denedinChapter1.Thefollowingtheorem,derivedinL iuandTsokos(2002), willhelpusshowthattheoptimalbandwidthforPDFisnotopt imalforCDF.This resultwillbeimportantinChapter3whenweexaminethereli abilityfunctionofthe Gumbeldistributionandtheconceptoftargettimeunderdes iredreliability. Theorem2.4.1. LetF(x)bethecdfoff(x)andassumef(x)possessesthesecon d derivative.Then AMISE ( ^ F ( x ))= 1 4 h 4 k 2 2 Z + 1 1 f 0 2 ( x ) dx + 1 n Z + 1 1 F ( x )(1 F ( x )) dx Proof. Integratingtheexpressionforthebiasof ^ f ( x )givenby(2.3.4)weobtain bias ( ^ F ( x ))= E ^ F ( x ) F ( x )= 1 2 h 2 k 2 f 0 ( x ) : (2.4.1) AlsoNadaraya(1964)derived Var ^ F ( x )= 1 n F ( x )(1 F ( x ))+ o ( 1 n )(2.4.2) Combiningthetwoexpressionsaboveandintegratingoverth ereallinewegetthe expressionfor AMISE ( ^ F ( x )) Sincethekerneldensityestimateofthereliabilityfuncti onisgivenby ^ R ( x )=1 ^ F ( x ) 22

PAGE 35

wemaymakethefollowinginferenceaboutthebiasandvarian ceof ^ R ( x ): bias ( ^ R ( x ))= E ^ R ( x ) R ( x )= 1 2 h 2 k 2 f 0 ( x ) : (2.4.3) and Var ^ R ( x )= 1 n R ( x )(1 R ( x ))+ o ( 1 n )(2.4.4) RecallthattheoptimalbandwidthestimateforPDFisgivenb y h optimal =[ C ( K ) k 2 2 C ( f 00 ) ] 1 5 n 1 5 Since AMISE ( ^ F ( x ))canbewrittenas AMISE ( ^ F ( x ))= 1 4 h 4 k 2 2 Z + 1 1 f 0 2 ( x ) dx + 1 n Z + 1 1 F ( x )(1 F ( x )) dx uponcloserstudyoftheexpressionfor AMISE ( ^ F ( x ))wemayconcludethefollowing: (i) Whenhischosensothat n 1 4 !1 ,thebiaspartdominates,thatis AMISE ( ^ F ( x ))= 1 4 h 4 k 2 2 Z + 1 1 f 0 2 ( x ) dx andn AMISE ( ^ F ( x )) !1 (ii) Whenthebandwidthhissmallsothath n 1 4 0,thevariancepartdominates, thus AMISE ( ^ F ( x ))= 1 n Z + 1 1 F ( x )(1 F ( x )) dx Also,ifweleth=a n 1 4 AMISE ( ^ F ( x ))= 1 n Z + 1 1 F ( x )(1 F ( x )) dx + m n ;m> 0 23

PAGE 36

so AMISE ( ^ F ( x ))attainsitsminimumvalue 1 n Z + 1 1 F ( x )(1 F ( x )) dx whenh=o( n 1 4 ) (iii) Itwasshownearlierthatforthekerneldensityestimate ^ f ( x )),theoptimal bandwidthis h optimal =[ C ( K ) k 2 2 C ( f 00 ) ] 1 5 n 1 5 Itisobviousthat h optimal doesnotsatisfythecondition h optimal n 1 4 0, so h optimal isnolongeroptimalforthekernelcdfestimate ^ F ( x ).Thesame argumentcanbeextendedtoconcludethat h optimal isnolongeroptimalforthe kernelestimateofthereliabilityfunction ^ R ( x ) VisualInspectionProceduretoDetermineOptimalBandwidt hforCDF andReliabilityFunctionsThefollowingfourstepprocedurewasproposedbyLiuandTso kos(2002)asavisual inspectionmethodtoarriveattheoptimalvalueofthebandw idthhforagivenkernel estimateofcdf.Thisanimportantprocedurethatisrelevan ttothekerneldensity estimationofthecumulativedistributionfunctionF(x)an dthereliabilityfunction R(x)Wewilltestitseectivenessontheestimatesofthecum ulativedistribution functionandthereliabilityfunctionfortheGumbeldistri butionpresentedinthe nextchapter. (i)Chooseapositivenumber h 1 andanintegerk. (ii)Forh= ih 1 k ,i=1,2,...kcalculatethecorresponding ^ R n anddisplaytheir graphs. (iii)Ifthesekgraphslookalmostthesame,chooseabigger h 1 andgobacktostep 1. 24

PAGE 37

(iv)Find i suchthatthegraphsbefore i lookverysimilarandthegraphsafter i lookquitedierentfrombefore. (v)Chooseanyh= ih 1 k ,i 0 Ashincreasesevenfurthersothath n 1 4 !1 ,thebiastermdominatesandwehave AMISE ( ^ F ( x ))= 1 4 h 4 k 2 2 Z + 1 1 f 0 2 ( x ) dx sothat nAMISE ( ^ F ( x )) !1 andtheminimumisattainedforh=o( n 1 4 ).This meansthatwhenhvariesfrom0towardthepositivedirection ,AMISE( ^ F n )stays almostthesameatthevalueofO( n 1 ).So,duringthisstagetheestimates ^ F n will lookverysimilar.Afterhexceedsacertainvalue,AMISE( ^ F n )willincreaserapidly atarateofO( h 4 )andthenewestimatesof ^ F n willdeviatefromthetrueF(x)quite signicantly.Intermsofhowthepilotbandwidthestimates houldbechosen,the optimalbandwidthforPDFisanaturalchoicesinceitisalwa yshigherthanthe optimalbandwidthforcdf.Themostintuitivechoiceforthe pilotbandwidthunder theassumptionofnormality,is h pilot =1 : 06 min (^ ; ^ IQR 1 : 34 ) n 1 = 5 25

PAGE 38

where^ istheestimateofthesamplestandarddeviationand ^ IQR istheinterquartile range(Silverman1986).Incaseoflongtaileddistribution andpossibleoutliers,a robustestimateof^ ismorepreferableandisusuallytakentobe ^ = median ( j x i ^ m j ) 0 : 6745 where^ m denotesthesamplemedian(Hogg1979)InordertostudyhowSi lverman's rankingofkernelsmayhavechangedunderthenewoptimalban dwidth,weperformed anextensivenumericalstudyusingtheGumbelprobabilityd istribution.Thestudy wasconductedinthefollowingmanner: (i)Wesimulatedm(m=50,100,200)Gumbeldistributionloca tionparameters fromtheuniformdistribution. (ii)Inordertoseewhateectstheincreaseofvariancehaso nourestimates,welet thescaleparameteroftheGumbeldistribution equalto1,2and4respectively. (iii)Usingtheobtainedmpairsof and ,wegeneratedn(n=15,30,50, 100,200)observationsfromtheGumbelPDFandobtainedreli abilityestimates underSilverman'sandLiu'soptimalbandwidthsandallseve nkernels. (iv)Forcomparisonpurposes,wecalculatedAMISEbetweent hetruereliability andthecorrespondingSilverman'sandLiu'sestimates,and rankedthekernels accordingtoitssize. Theschematicdiagramofournumericalstudyispresentedin gure2.8.Theresults intables2.3-2.5representtheaverageAMISEacrossallsam plesofsizenandfor varyingvaluesof fortheaveragevaluesofthepilotbandwidth h pilot ,Silverman's bandwidth h opt ,andLiu-Tsokosoptimalbandwidth h .Specically,wecalculated theestimatedAMISEacrossallsamplesnwhere AMISE ( R n ( t ) ;R ( t ))= N X i =1 Z ( R n ( t ) R ( t )) 2 dt N ; 26

PAGE 39

Figure2.8:NumericalStudyofKernelRanking acrossallfailuretimestforthereliabilityfunctionR(t) ,withNbeingthenumber ofsimulations.Withrespecttotheevaluationoftwoselect edoptimalbandwidths, namelySilverman's h opt andLiu-Tsokos's h ,asafunctionoftheselectedkernels, theLiu-Tsokosoptimalbandwidthresultsinasmallermeani ntegratedsquareer27

PAGE 40

ror.UsingLiu-Tsokosbandwidththeeectivenessofallker nelswasevaluated,with thetopthreekernelsbeingtheEpanechnikov,CosineandGau ssian.Tables2.3-2.5 suggestthatforsamplesizen > 100weobtainapproximatelythesameprobability densityfunctionsubjecttothesamebandwidth.SincetheGa ussiankerneloers severalanalyticadvantagesweconcludethatGaussiankern elshouldbeused.Furthermore,theGaussiankernelseemstobemorestableaswein creasethevariance. Forsamplesizen < 100wehavefoundthatEpanechnikovkerneldensityfunction inconjuctionwithoptimalbandwidthwillgivethebestesti mates.Theincreasein variance hadnoeectontherankingofourkernels.Nextweproceedtos tudy Kernel n h opt h pilot h MISE ( h opt ) MISE ( h pilot ) MISE ( h ) Epanechn. 15 1.00 0.5762 0.95 0.0026 0.0068 0.0026 Cosine 15 1.028 0.576 0.95 0.0027 0.0072 0.0027 Gaussian 15 0.4517 0.5762 0.95 0.0029 0.009 0.008 Epanechn. 30 0.8706 0.3519 0.75 0.013 0.026 0.015 Cosine 30 0.8946 0.3519 0.75 0.013 0.026 0.014 Gaussian 30 0.3932 0.3519 0.75 0.014 0.029 0.019 Epanechn. 50 0.786 0.4121 1.5 0.0025 0.003 0.002 Cosine 50 0.8077 0.4121 1.5 0.0025 0.003 0.0019 Gaussian 50 0.355 0.4121 1.5 0.0025 0.0018 0.0020 Epanechn. 100 0.786 0.4121 1.5 0.0013 0.003 0.002 Cosine 100 0.8077 0.4121 1.5 0.0013 0.003 0.0019 Gaussian 100 0.355 0.4121 1.5 0.0012 0.0018 0.001 Epanechn. 200 0.786 0.4121 1.5 0.0013 0.003 0.002 Cosine 200 0.8077 0.4121 1.5 0.0013 0.003 0.0019 Gaussian 200 0.355 0.4121 1.5 0.0012 0.008 0.0018 Table2.3:MeanIntegratedSquareError(MISE)fortheTopTh reeBandwidthsforData FromGumbel(n=15,30,50,100,200, =1) bandwidthrobustnessfortheoptimalbandwidth h underLiu'svestepprocedure. 28

PAGE 41

Kernel n h opt h pilot h MISE ( h opt ) MISE ( h pilot ) MISE ( h ) Epanechn. 15 1.00 0.9062 1.15 0.0522 0.06 0.053 Cosine 15 1.028 0.9062 1.15 0.052 0.06 0.053 Gaussian 15 0.4517 0.9062 1.15 0.090 0.110 0.081 Epanechn. 30 0.8706 0.789 1 0.1046 0.1061 0.102 Cosine 30 0.8946 0.789 1 0.105 0.1263 0.114 Gaussian 30 0.3932 0.789 1 0.1178 0.1282 0.141 Epanechn. 50 0.786 0.9179 1.35 0.0031 0.0028 0.0024 Cosine 50 0.786 0.9179 1.35 0.0032 0.0029 0.0024 Gaussian 50 0.786 0.9179 1.35 0.1407 0.1641 0.1311 Epanechn. 100 0.786 0.4121 1.5 0.0020 0.003 0.002 Cosine 100 0.8077 0.4121 1.5 0.0020 0.004 0.0019 Gaussian 100 0.355 0.4121 1.5 0.0042 0.0048 0.001 Epanechn. 200 0.786 0.4121 1.5 0.0013 0.004 0.002 Cosine 200 0.8077 0.4121 1.5 0.0013 0.003 0.0019 Gaussian 200 0.355 0.4121 1.5 0.0019 0.008 0.0005 Table2.4:MeanIntegratedSquareError(MISE)fortheTopTh reeBandwidthsforData FromGumbel(n=15,30,50,100,200, =2) 2.5BandwidthRobustness NowwewishtotakealookathowAMISEchangesasweincrementa llydecrease andincrease h optimal underLiu'sve-stepprocedureforeachkernel,thatiswish to studyhowrobusttheoptimalbandwidthforeachkernelisbyo bservingtherateof changeofAMISEastheoptimalbandwidthforeachkernelunif ormlydecreasesor increases.Inordertodothiswehaveusedthesamenumerical procedurerepresented bygure2.8exceptthatwestudiedhowsensitivethechangei nbandwidthselection iswithrespecttochangeinAMISEforaxedbandwidthforeac hofthethreekernels. TostudytherateofincreaseofAMISEwehaveusedtheratioof twoconsecutive 29

PAGE 42

Kernel n h opt h pilot h MISE ( h opt ) MISE ( h pilot ) MISE ( h ) Epanechn. 15 1.00 1.5892 1.75 0.148 0.139 0.121 Cosine 15 1.028 1.5892 1.75 0.148 0.15 0.13 Gaussian 15 0.4517 1.5892 1.75 0.16 0.19 0.17 Epanechn. 30 0.8706 1.554 1.5 0.058 0.065 0.044 Cosine 30 0.8946 1.554 1.5 0.057 0.066 0.042 Gaussian 30 0.3932 1.554 1.5 0.069 0.071 0.058 Epanechn. 50 0.786 1.995 2.25 0.012 0.018 0.009 Cosine 50 0.786 1.995 2.25 0.0011 0.0018 0.009 Gaussian 50 0.786 1.995 2.25 0.0011 0.018 0.011 Epanechn. 100 0.786 0.4121 1.5 0.0028 0.003 0.002 Cosine 100 0.8077 0.4121 1.5 0.0028 0.003 0.0019 Gaussian 100 0.355 0.4121 1.5 0.0032 0.0038 0.0018 Epanechn. 200 0.786 0.4121 1.5 0.0009 0.0011 0.0008 Cosine 200 0.8077 0.4121 1.5 0.0008 0.0002 0.0009 Gaussian 200 0.355 0.4121 1.5 0.001 0.0014 0.0007 Table2.5:MeanIntegratedSquareError(MISE)fortheTopTh reeBandwidthsforData FromGumbel(n=15,30,50,100,200, =4) AMISE'sdenedas AMISE rate = AMISE i AMISE i 1 ;i =1 ;:::;l (2.5.1) wherethesubscriptindicatesani-percentofthebandwidth increaseordecreasein bandwidth.TheresultsarepresentedinTables2.6-2.8.We xedthetopthreekernels, namelyEpanechnikov,CosineandGaussian,andstudiedtheb ehaviorofAMISE underthesamenumericalstudyasgivenbygure2.6.Theopti malbandwidth h wasvariedinbothdirectionsbyacertainpercentageandsam plesizen=20,50,and 100.Theresultsaregivenintables2.6-2.8andindicatetha tthesmallestincreasesin AMISEwerepresentundertheGaussianKernelaswexedtheke rnelandincreased 30

PAGE 43

thesamplesize.ThissuggeststhattheGaussiankernelismo restableasweincrease thesamplesize.Undertheoptimalbandwidthforeachkernel wehavefoundthatthe Bandwidth %Change Epanechnikov Cosine Gaussian h -99% 1.831 1.9121 1.781 h -90% 0.38860 0.40 0.2309 h -50% 0.1325 0.1358 0.0696 h 0.072 0.073 0.08 h +50% 0.0550 0.056 0.09 h +100% 0.0441 0.045 0.099 h +200% 0.029 0.03 0.0134 h +500% 0.0133 0.0136 0.023 h +1000% 0.0205 0.018 0.044 Table2.6:AMISERateofChangefor h ,n=20 Bandwidth %Change Epanechnikov Cosine Gaussian h -99% 2.328 2.786 0.9097 h -90% 0.1415 0.1656 0.0652 h -50% 0.025 0.0305 0.009 h 0.0097 0.012 0.004 h +50% 0.0058 0.007 0.0039 h +100% 0.0043 0.0496 0.0052 h +200% 0.0037 0.0037 0.0098 h +500% 0.0088 0.0066 0.0277 h +1000% 0.0252 0.0195 0.049 Table2.7:AMISERateofChangefor h ,n=50 rateofincreaseinAMISEforthetopthreekernelsisvirtual lyidentical,whichleads toconcludethatanincreasein h foragivenkernelwillproduceroughlythesame percentageincreaseinAMISEandthattheoptimalbandwidth foraparticularkernel 31

PAGE 44

isrobustwithrespecttothechoiceofkernel.Asexpectedas nincreases,AMISE decreases.WealsonoticethatAMISEishigherifwechooseam uchlowervalue for h ratherthanahighervalue,sothepenaltyofoverestimating thebandwidthis lowerthanthepenaltyforunderestimating.Also,AMISEdoe snotincreasemuchas weincreasethebandwidth.Fromtable2.5weseethatifweinc rease h tenfold, AMISEis"only"0.0415forEpanecnikov,0.0399forCosinean d0.048fortheGaussian kernel.Weknowthatifthebandwidthincreasesweruntheris kofoversmoothing whichmayhideimportantfeaturesofthedata,suchasbimoda lity,andcauseusto drawwrongconclusions.Itshouldbenotedthatoursimulati onhasindicatedthat lim i !1 AMISE rate =1. Bandwidth %Change Epanechnikov Cosine Gaussian h -99% 0.5234 0.5411 0.381 h -90% 0.0404 0.042 0.031 h -50% 0.0073 0.0076 0.0042 h 0.0021 0.0022 0.0022 h +50% 0.0023 0.0023 0.0031 h +100% 0.0029 0.0029 0.0047 h +200% 0.0056 0.0053 0.0095 h +500% 0.0198 0.0187 0.027 h +1000% 0.0415 0.0399 0.048 Table2.8:AMISERateofChangefor h ,n=100 32

PAGE 45

2.6Conclusion Inthischapterwestudiedallsevensymmetrickernelsusedi nkerneldensityestimationandappliedtheasymptoticmeansquareerrortorankt hetopthreekernels dierentfromSilverman(1986).OurrankingplacestheGaus siankernelthird,after EpanechnikovandCosinekernels.Thiswasdoneunderasimpl evisualprocedure introducedbyLiuandTsokos(2002),whichproducesanoptim alcdfandreliability kerneldensityestimatesandistobeusedintherestofourst udy.Weshowedthat samplesizenisimportantinselectingtheappropriatekern elfunction.Generally, theEpanechnikovkernelperformsbetterforsamplesizen < 100,whileallthree areroughlythesameforn 100.SincetheGaussiankerneloersanalyticand numericsimplicity,werecommendtheGaussiankernelbeuse dforn 100.Wealso showedthat h optimal forthetopthreekernelsisrobustwithrespecttothechoice of thekernel,aswellasthesamplesizen.Thendingsinourpre sentchapterwillbe utilizedinreliabilityandthemodelingofroods. 33

PAGE 46

3Ordinary,Bayes,EmpiricalBayes,andKernelDensityReliabi lity EstimatesfortheGumbelFailureModel 3.1Introduction Extremevalueprobabilitydistributionshavebeenusedee ctivelytomodelvarious problemsinengineering,environment,business,etc.Some keyreferencesareBurton andMarkopoulos(1985),Naess(1998),Osellaet.al.(1992) ,Ramachandran(1982) Raoetal.(1997),SastryandPi(1991),Silbergleit(1996), SuzukiandOzaha(1994), Tsokos(1999),andYue(2000).ArecentbookbyKotzandNadar ajah(2000)lists overftyapplications,rangingfromacceleratedlifetest ingtoearthquakes,roods, horseracing,rainfall,queuesinsupermarkets,seacurren ts,windspeeds,andrace trackrecords.Tsokos(1999)analyzedamodiedextremeval uedistributionand derivedtheminimumvarianceunbiased(MVU)andBayesianes timatesofthereliabilityfunctionunderthegeneraluniform,exponential,a ndinvertedgammapriors, andthemeansquareerrorlossfunction.Usinghisworkasafo undation,theobjective ofthisstudyistomodifytheclassicalGumbel,ordoubleexp onential,probability distributiontocharacterizethefailuretimesofagivensy stem.Themodicationis necessaryinordertoobtainanalyticallytractableestima tesofthedesiredfunctions andtoensurethattimetofailurecanbeconsideredmodiedt oreliabilityanalysis. WeareinterestedinobtainingordinaryandBayesianestima tesoftheGumbelreliability,failurerate,andcumulativefailureratefunctio ns.Inadditiontoobtaining maximumlikelihood(ML)andMVUestimates,wehavedevelope dthesubjectmodel inBayesandempiricalBayessettings.Wearealsointereste dinobtainingordinary, Bayes,andempiricalBayesestimatesofthetargettimet c subjecttoadesiredand 34

PAGE 47

speciedreliability.Thatis,wewanttoknowwhatthetimet ofailuret c iswith atleast(1c )100%assurance.Forexample,wewanttobeatleast95%certa in thatthesystemwillbeoperabletotimet 0 : 05 .IntheBayesiansetting,weusethe naturalconjugatepriorunderthemeansquareerrorlossfun ction.Lindley'sapproximationprocedureisusedtoobtainnumericalresultsthati llustratetheusefulnessof thestudy.Finally,weassumethefailuredatadoesnottthe Gumbelprobability distribution,andconsistentwithourresultsinchapter2, obtainthekerneldensity estimatesfortheGumbelreliability.Inadditiontoanalyt icalresults,wehaveconductedanextensivenumericalsimulationinordertoillust ratetheusefulnessofthe inferenceproceduresdiscussed.Insummary,afterintrodu cingthemodiedGumbel failuremodelinsection3.2,weaimtoaccomplishthefollow ing: (i)Insection3.3,presenttheMLandMVUestimatesofreliab ility,failurerate, cumulativefailureratefunctions,andthetargettimesubj ecttoaspecied reliability t c (ii)Insection3.4,derivetheBayesianandempiricalBayes estimatesofreliability, failurerate,cumulativefailureratefunctions,andtheta rgettime t c ,underthe naturalconjugatepriorandasquareerrorlossfunction. (iii)Insection3.5,presentthenon-parametrickernelden sityestimatesofthefunctionsunderstudy. (iv)Insection3.6,performanextensivenumericalanalysi stocomparetheestimates andillustratetheusefulnessofthemethodology.Inthisse ction,wewillusethe vestepprocedureintroducedinchapter2toobtaintheopti malkerneldensity estimatesofreliability,failurerate,cumulativefailur erate,andtargettime t c toascertainhowwellthekerneldensityestimatesperformw hencomparedwith theirparametriccounterpartsinbothordinaryandBayesia nsettings. (v)Insection3.7,presentconcludingremarksandrecommen dations. 35

PAGE 48

3.2TheGumbelFailureModel FortheGumbelfailuremodel,theprobabilitydistribution function(PDF)andthe cumulativedistributionfunction(CDF)ofthefailuretime attimetaregiven,respectively,by f ( t ; ; )= 1 e t e t ; 1 0(3.2.1) and F ( t ; ; )= Pr ( X t )= e e t ; (3.2.2) where and arethelocationandscaleparameters.Thelikelihoodfunct ionL( ),isgivenby L ( t ; ; )= n exp f n X i =1 t i n X i =1 exp ( t i ) g : (3.2.3) TheGumbelfailuremodelhasbeenusedinreprotection,ins uranceproblems,predictionofearthquakemagnitudes,carbondioxidelevelsin theatmosphere,andhigh returnlevelsofwindspeedsinthedesignofstructuresamon gothers.Basedonrecord values,Ahsanullah(1990,1991)obtainedthemaximumlikel ihood(ML),bestlinear invariant(BLI)andminimumvarianceunbiased(MVU)estima torsoftheGumbel locationandscaleparameters,andAliMousaetal.(2001)ob tainedtheBayesian estimatorsofthesameunderJerey'snon-informativeprio r.Inthepresentstudy, weshallmodifythesubjectmodelandapplyitinreliability inordinary,Bayesian andempiricalBayessettings.Inthenextsection,weconsid ertheordinaryestimators,namelythemaximumlikelihood(ML)andminimumvarian ceunbiased(MVU) estimators.Toourknowledge,theMVUestimatorshavenotbe enderived,sothe derivationispresented. 36

PAGE 49

3.3ReliabilityModeling Let t 1 t 2 ,..., t n bethefailuretimesthatfollowtheGumbelPDFgivenby(3.2. 1). Thereliabilityattimetofasystemwhoselifefollowsthepr obabilitylawf(x; )is givenby R ( t ; )= Z 1 t f ( x ; ) dx =1 F ( t )(3.3.1) Underthesamereliabilityfunction R ( t ; ),thefailurerateandcumulativefailure ratefunctionsaregivenby h ( t )= @R ( t ; ) @t R ( t ; ) (3.3.2) and H ( t )= ln ( R ( t ; )) : (3.3.3) Thefailureratefunctioncanbeinterpretedastheprobabil ity,perunitoftime,that theitemwillfailaftertimet,giventhattheitemhasoperat eduptotimet.Inother words,thefailureratefunctioncanbeinterpretedasthepr obabilityofinstantaneous failure,giventhattheitemhasoperateduptotimet.Ifwelet g ( t ; )= e e t and = e ; thentheCDFandreliabilityfunctionsoftheclassicalGumb elfailuremodelcanbe writtenas F ( t ; )=[ g ( t )] (3.3.4) and R ( t ; )=1 [ g ( t )] : (3.3.5) Notethatg(t)ismonotoneincreasingand R ( t ; )isboundedfromaboveandbelow. Inadditiontoparameterthescale > 0,thenewparameter isalsopositive. 37

PAGE 50

Sincefailuretimesarepositivequantities,thiswillallo wforbetterconsistencyofparameters.Usingthenewparameterization,theprobability densityandthelikelihood functionscanbewrittenas f ( t ; )= @F ( t ; ) @ =[ g ( t )] ( 1) [ g ( t )] 0 (3.3.6) and L ( t ; )= n g 0 ( t i )[ g ( t i )] 1 : (3.3.7) Thefailurerateandcumulativefailureratefunctionsarer espectivelygivenby h ( t; )= g 0 ( t )[ g ( t )] g ( t )(1 g ( t ) ) (3.3.8) and H ( t; )= ln (1 [ g ( t )] ) : (3.3.9) Bytakingthenaturallogarithmofbothsidesofequation(3. 3.4)andsolvingfort,we obtaintheexpressionforthetargettime t c underthedesiredreliability(1c )100% givenby t c =( ln ( ln ( c ) )) : (3.3.10) Themaximumlikelihoodestimates(MLE)for and canbederivedfromequations (3.2.3)and(3.3.7)bysolving @lnL @ =0and @lnL @ =0,fromwhichweobtaintheML estimates ^ ML + t i e t i = ^ e t i = ^ = t (3.3.11) and ^ ML = n G ; (3.3.12) where G = n X i =1 lng ( t i ) : (3.3.13) 38

PAGE 51

Equations(3.3.11)and(3.3.12)arenotanalyticallytract ableandmustbesolved numericallytoobtainapproximateMLE'sof and .Bytheinvarianceproperty oftheMLE's,wecanobtaintheMLestimatesofthesubjectfun ctionsbyreplacing parameters and withtheirMLestimates^ and ^ Incomplexsystems,thecumulativeeectofbiasmightbequi teconsiderableanda systemmightproveunsatisfactoryduringoperationtime.F orthisreason,wederive theminimumvarianceunbiased(MVU)estimatorsofthefunct ionsforthemodied Gumbelfailuremodelandapplythemtoreliabilityanalysis .Inordertoderivethe MVUestimators,weneedtondasucientandcompletestatis ticfor andnd itsdistribution.Givenasampleofsizenoffailuretimes, t 1 t 2 ,..., t n ,andthe cumulativedistributionfunctionin(3.3.4)bytheNeymann factorizationtheoremand usingthelikelihoodfunctiongivenby(3.3.7),itiseasily shownthatthequantityG =P ni =1 lng ( t i )issucientandcompletefor .TondthedistributionofG,weuse thecharacteristicfunctionargument.Ifweletp(t)=-ln(g (t)),thenthecharacteristic functionofp(t)is ( w )= E ( e itp ( t ) )= Z 1 0 e wp ( t ) g 0 ( t ) g ( t ) 1 ds or ( w )= Z 1 0 e ws e s ds = w ; whichisthecharacteristicfunctionfortheexponentialra ndomvariable.Usingthe propertythattheGammaprobabilitydistributionrepresen tsthesumofnexponentiallydistributedrandomvariables,weconcludethatGisd istributedasaGamma randomvariablewithparameters =nand = .Itsprobabilitydensityfunction isgivenby f ( G ; n; )= G n 1 ( n ) n e G ; lng ( t )
PAGE 52

estimateofthereliabilityfunctionR(t)isgivenby ^ R MVU ( t )=1 (1+ lng ( t ) G ) n 1 ; lng ( t )
PAGE 53

Recallthatgivenasampleofsizenoffailuretimes, t 1 t 2 ,..., t n ,themodied Gumbelreliabilityfunctionisgivenby R ( t ; )=1 [ g ( t )] where g ( t ; )= e e t and ( ; )= e : Weassumethattheparameter isxedandthattheparameter behavesasa randomvariable.Thisimpliesthatthelocationparameter oftheclassicalGumbel probabilitydistributionbehavesasarandomvariable.Int heprevioussection,we showedthatstatistic G = n X i =1 lng ( t i ) issucientandcompletefor andisdistributedasaGammarandomvariable. Therefore,wearejustiedinassumingthatthenaturalconj ugatepriorofparameter followstheGammaprobabilitydistributionwithparameter s and ,sothatthe naturalconjugatepriorhastheform g ( ; ; )= ( ) 1 e : ThejointPDFof t 1 ,..., t n isgivenby L ( t ; )= Z f ( t j ) g ( ; ; ) d L ( t ; )= Z 1 0 ( ) n + 1 e g 0 ( t i )[ g ( t i )] 1 d; whichgives L ( t ; )= ( n + ) ( )( + G ) n + g 0 ( t ) g ( t ) : (3.4.1) 41

PAGE 54

Similarly,theposteriordistributionof isgivenby f ( j t )= L ( t ; ) g ( ) R L ( t ; ) g ( ) d or f ( j t )= ( + G ) n + ( n + ) n + 1 e ( + G ) (3.4.2) Therefore,theBayesianestimatesofR(t),h(t),H(t),and t c underthesquareerror lossare: ^ R B ( t )=1 Z 1 0 [ g ( t )] ( + G ) n + ( n + ) n + 1 e ( + G ) d or ^ R B ( t )=1 ( + G + G lng ( t ) ) n + (3.4.3) ^ h B ( t )= g 0 ( t ) g ( t ) Z 1 0 [ g ( t )] g ( t )(1 g ( t ) ) ( + G ) n + ( n + ) n + 1 e ( + G ) d ^h B ( t )= R 1 0 u ( ) e ( + G )+( n + 1) ln d R 1 0 e ( + G )+( n + 1) ln d (3.4.4) for u ( )= g 0 ( t )[ g ( t )] g ( t )(1 g ( t ) ) ^ H B ( t )= ln ( ^ R B ( t ))(3.4.5) ^ t B = Z 1 0 ( ln ln ( ln ( c )) ( + G ) n + ( n + ) n + 1 e ( + G ) d ^ t B = (( n + ) ln ( + G ) ln ( ln ( ))) ; (3.4.6) where ( x )= dln ( x ) dx isthedi-gammafunction.TheBayesianestimate ^h B hastobeapproximatedusing theLindleyapproximationmethod.Weshallbrieryoutlinet hemethodanddiscuss itsusefulnessinthepresentstudy. 42

PAGE 55

LindleyApproximationof ^ h B ( t ) WhentheBayesianestimatesarenotinclosedform,theLindl eyapproximation methodallowsforevaluationoftheratioofintegralsofthe form I ( x 1 :::x n )= R u ( ) e L ( )+ ( ) d R e L ( )+ ( ) d (3.4.7) where L ( )= log l ( x j ) denotesthelogoflikelihoodfunction,and ( )= log ( g ( )) denotesthelogofthepriordensityand u ( )isanarbitraryfunctionof .The approximationisbasedontheresultthatforasucientlyla rgen,sothatL( )dened aboveconcentratesaroundauniquemaximumlikelihoodesti mator ^ = ^ ( x 1 :::x n ), I(.)canbeexpressedapproximatelyas I ( : ) u ( )+0 : 5 p X i =1 p X j =1 f @ 2 u ( ) @ i @ j +2 @u ( ) @ i @ ( ) @ j g ^ ij +0 : 5 p X i =1 p X j =1 p X k =1 p X l =1 f @ 3 L ( ) @ i @ j @ l @u ( ) @ k g ^ ij ^ kl for px 1 ,where ij denotesthe(i,j)element ^ 1 = ^ = ^ ij and ^ ij = @ 2 L ( ) @ i j ; allevaluatedatthemaximumlikelihoodestimator = ^ .Theaboveformulationof theLindleyapproximationisforestimatingtheposteriord istributionof u ( )given thelikelihoodfunction L ( )andprior g ( ).OurapplicationoftheLindleyapprox43

PAGE 56

imationwillvaryslightlysinceweknowtheposteriordistr ibutionof .Let q ( )= ( + G )+( n + 1) ln SinceweareusingLindleyapproximationtocalculatetheBa yesianestimateofthe failureratefunction ^h B givenby(3.4.4),forp=1weget I ( : ) u ( ^ ) 0 : 5 1 @ 2 q=@ 2 ( @ 2 u @ 2 ( @u=@ )( @ 3 q=@ 3 ) @ 2 q=@ 2 )(3.4.8) evaluatedat = n + 1 + G andthesolutionof @q @ =0 InordertoderiveanexpressionfortheLindleyApproximati onof ^ h B ,recallthat h ( t )= u ( )= g 0 ( t )[ g ( t )] g ( t )(1 g ( t ) ) : Alsoitcanbeshownthat @u @ = u ( ) c ( ) ;c ( )= 1 + lng ( t ) 1 g ( t ) and @ 2 u @ 2 = u ( ) c 2 ( ) u ( ) 2 + u ( ) g ( t ) [ lng ( t ) 1 g ( t ) ] 2 : Theexpressionfor ^h B is ^ h B ( t ) h ( t )+ h ( t ) n + 1 ( 1 +3 lng ( t ) 1 g ( t ) + [ lng ( t ) 1 g ( t ) ] 2 (1+ g ( t ) )(3.4.9) evaluatedat ^ = n + 1 + G 44

PAGE 57

Next,weproceedtoderivetheempiricalBayesreliabilitye stimates. EmpiricalBayesEstimates EmpiricalBayesestimationwasintroducedbyRobbins(1980 ).Itparallelsthe Bayesianestimationphilosophyexceptthatthepriorproba bilitydistributionisunknownandnotassumed.Itassumesthattherealizationsofth eunderlyingfailure modelparameterhavebeenestimatedseveraltimesbefore.T herefore,theestimates basedonpastinformationhelpusconstructthepriorprobab ilitydistributionempirically.Considerthesituationwherewehavekindepende ntrandomfailuretimes T 1 ;T 2 ;:::;T k withthesameprobabilitydensityfunction dF ( t j ),andeachofthem havingnrealizations: T 1 : t 11 ;t 21 ;:::;t n 1 T 2 : t 12 ;t 22 ;:::;t n 2 :: T k : t 1 k ;t 2 k ;:::;t nk Weshowedintheprevioussectionthatthenaturalconjugate priorofparameter wastheGammaprobabilitydistributionfunctionwithparam eters and g ( ; ; )= ( ) 1 e Inthisstudy,theempiricalBayesestimateswillconsistof estimating and for thenaturalconjugatepriorfromkpastsamples.Tothisend, weassumethatwe havekpastsamplesoffailuretimesofsizenandacurrentsam plealsoofsizen.If wedene G j = lng ( t j ) forj=1...k+1,thatisforeachofthek+1samples,thenitfol lowsthat m j = n G j followstheinvertedgammaprobabilitydistributionsince GfollowstheGamma 45

PAGE 58

probabilitydistribution.Morespecically,theposterio rdistributionof isgivenby f ( m j )= ( n ) n e n m ( n ) m n +1 : (3.4.10) Thereforethemarginaldistributionofmis f ( m )= Z f ( m j ) g ( ; ; ) d = n n m 1 B ( ;n )( n + m ) n + (3.4.11) whereB( ,n)istheBetafunctiondenedas B ( t ; ;n )= Z 1 0 t 1 (1+ t ) n dt Fromtheabovewecanshowthatthemomentsofmaregivenby: E ( m j )= n ( n 1) (3.4.12) and E ( m 2j )= n 2 ( +1) ( n 1)( n 2) 2 (3.4.13) Usingthemethodofmomentsitfollowsthattheestimatesof and are ^ = P 2 Q P 2 (3.4.14) ^ = P Q P 2 (3.4.15) where P = n 1 kn kj =1 m j andQ = ( n 1)( n 2) kn 2 kj =1 m 2j WeobtainthecorrespondingEmpiricalBayesestimatesbypl uggingin^ ^ and = n +^ 1 ^ + G k +1 fortheBayesianestimates ^ R B ( t ), ^ h B ( t ), ^ H B ( t )and ^t B .Inthefollowingbrief 46

PAGE 59

sectionweoutlinethestepsthatareusedtocalculatenon-p arametrickerneldensity estimatesofreliability. 3.5Non-parametricKernelDensityEstimatesofReliabilit y For t 1 t 2 ,..., t n i.i.d.randomvariableshavingacommonprobabilitydensit yfunction f(t),thekerneldensityestimateofreliabilityisdenedb y ^ R n ( t )=1 1 nh n X i =1 Z t 1 K ( y T i h ) dy: (3.5.1) Bysolvingforthequantile t forwhich 1 nh n X i =1 Z t 1 K ( y T i h ) dy = ; (3.5.2) wegetthenonparametricestimateof"targettime"tospeci edreliability(1)100%. Likewise,thenonparametricestimateofthefailureratean dcumulativefailurerate functionsaregivenby: ^ h n ( t )= @ ^ R n ( t ) @t ^ R n ( t ) (3.5.3) and ^ H n ( t )= ln ( ^ R n ( t )) : (3.5.4) Itshouldbeemphasizedagainthatthenon-parametricappro achshouldnotbeutilizedunlessweareunabletoidentifyawelldenedparametr icprobabilitydistribution functiontomodelgivendata.Non-parametrickerneldensit yestimatesofreliability, failurerate,cumulativefailurerate,andtargettimesubj ecttospeciedreliability arecalculatedusingthevestepprocedurefromchapter2,w hichgivesusthevalue oftheoptimalbandwidth h opt : (i)Chooseapositivenumber h 1 andanintegerk. (ii)Forh= ih 1 k ,i=1,2,...kcalculatethecorresponding ^ R n anddisplaytheir 47

PAGE 60

graphs. (iii)Ifthesekgraphslookalmostthesame,chooseabigger h 1 andgobacktostep 1. (iv)Find i suchthatthegraphsbefore i lookverysimilarandthegraphsafter i lookquitedierentfrombefore. (v)Chooseanyh= ih 1 k ,i
PAGE 61

whereNisthenumberofsimulationsandnsamplesize.Theext ensivenumerical simulationwasconductedinthefollowingmanner: (i)Undertheassumptionthatthenaturalconjugatepriorfo rparameter isthe Gammaprobabilitydistributionandinordertoascertainth einruencethevarianceandsamplesizeoftheconjugatepriorhaveonthemodie dGumbelreliabilityestimates,wegeneratedk=5,10,15valuesof fromtheGamma probabilitydistributionwith( =5, =0.5)forsmallvariance,( =5, =1)formediumvariance,and( =5, =2)forlargevariance.Thevalues aregivenintables3.1-3.3andthegraphsofthethreedistri butionsaregiven respectivelybyFigures3.1-3.3. 1.702538 3.944660 1.173485 1.122669 2.984586 1.199388 4.556971 3.248614 2.566356 3.817823 2.327332 3.447440 4.679420 2.073639 5.699439 Table3.1:Generated ValuesUndertheGammaPriorWith =5, =0.5 11.780646 4.026650 3.027582 4.544242 11.630842 8.064418 9.020463 10.746460 3.146756 4.677712 3.656309 5.042761 3.412835 4.841834 7.024123 Table3.2:Generated ValuesUndertheGammaPriorWith =5, =1 10.178833 23.524719 11.276882 6.016636 25.579815 10.967756 29.885688 52.856968 22.986660 31.397815 9.508822 21.666094 37.301717 22.154025 13.828831 Table3.3:Generated ValuesUndertheGammaPriorWith =5, =4 49

PAGE 62

051015202530 0.000.020.040.06 ThetaGamma(alpha = 5, beta =0.5) Figure3.1:GammaPriorfor =5, =0.5 051015202530 0.000.050.100.150.20 ThetaGamma(alpha = 5, beta =1) Figure3.2:GammaPriorfor =5, =1 (ii)Toascertaintheinruenceofsamplesizeonourestimate s,wegeneratedsamples ofsizen(n=20,50,100,200)fromtheGumbelprobabilitydis tributionwith kparameters fromthepreviousstepandscaleparameter =1forsmall variance, =2formediumvariance,and =4forlargevariance,inorderto establishwhatinruencetheGumbelvariancehasontherelia bilityestimates. 50

PAGE 63

051015202530 0.00.10.20.30.4 ThetaGamma(alpha = 5, beta =4) Figure3.3:GammaPriorfor =5, =4 (iii)Foreachsampleofsizen(n=20,50,100,200),kernelde nsityestimateswere derivedforeachfunctionusingthevestepprocedurefromc hapter2.Consistentwiththeresultsfromchapter2,forn < 100weusedtheEpanechnikov kernelandforn 100theGaussiankernel.Hereweillustratetheprocessof ndingtheoptimalReliabilityestimateusingtheve-step procedureforn= 50: (a) h pilot =1.5andchooseintegerk=7. (b)Forh= ih 1 k ,i=1,2,...7calculatethecorresponding ^ R n anddisplay theirgraphs. (c)Ifthesekgraphslookalmostthesame,chooseabigger h 1 andgobackto step1. (d)For i =2and i =4thegraphslookquitedierent(Figure3) (e)Chooseanyh= ih 1 k ,i
PAGE 64

0.4 15 0.2 0 10 5 0 1 25 0.8 20 0.6 OptRel (h=0.4286) Reliab(h=0.86) Figure3.4:OptimalReliabilityforh=0.4286 (iv)Eachofthemaximumlikelihood,minimumvarianceunbia sed,Bayes,Empirical BayesandKernelDensityestimateswereobtainedforthefun ctionsunderstudy andusingIntegratedMeanSquareErrorcomparedwiththeval uesofthetrue functions. Aschematicdiagramofthecompletestepbystepprocessofth enumericalanalysis ispresentedinFigure3.5. 52

PAGE 65

Figure3.5:NumericalStudyofGumbelReliability 53

PAGE 66

Tables3.4-3.7summarizethevaluesofMISEbetweenthesimu latedtruereliability,truefailurerate,truecumulativefailurerateandtru etargettimefunctionson onesideandtheirmaximumlikelihood(ML),minimumvarianc eunbiased(MVU), Bayes,empiricalBayes(EB),andkerneldensity(KD)estima tesontheother,for thescaleparametervalues =1(smallvariance), =2(mediumvariance)and =4(largevariance)andn=20,50,100,200.Aswecanseefromt ables3.4-3.7, theconsistentclosestestimateswereBayes,followedbyem piricalBayes,minimum varianceunbiased,maximumlikelihoodandkerneldensity. Asthesamplesizen increased,thevaluesofMISEapproachedzeroforalltheest imates,indicatingthat theywereasymptoticallyecient.Thekerneldensityestim atesdidnotperformwell whencomparedwiththeirparametriccounterpartsforlarge valuesofGumbelvariance.Wehavealsofoundthattheincreaseinvarianceinthep riordidnothave anyinruenceontherankingofourreliabilityestimates;ho wever,theincreaseinthe GumbelvariancedecreasedtheeectivenessoftheKDE's. 54

PAGE 67

n, =1 20 50 100 200 MISE(R(t), ^ R ML ) 0.1795 0.007 0.0106 0.0113 MISE(R(t), ^ R MVU ) 0.024 0.0002 0.023 0.0111 MISE(R(t), ^ R Bayes (t)) 0.0024 0.0002 0.0044 0.0004 MISE(R(t), ^ R EB (t)) 0.0027 0.0014 0.0234 0.0011 MISE(R(t), ^ R KD (t)) 0.153 0.022 0.0084 0.001 n, =2 20 50 100 200 MISE(R(t), ^ R ML ) 0.09 0.007 0.09 0.011 MISE(R(t), ^ R MVU ) 0.025 0.0002 0.04 0.001 MISE(R(t), ^ R Bayes (t)) 0.0021 0.001 0.001 0.0003 MISE(R(t), ^ R EB (t)) 0.013 0.0014 0.008 0.001 MISE(R(t), ^ R KD (t)) 0.132 0.022 0.012 0.01 n, =4 20 50 100 200 MISE(R(t), ^ R ML ) 0.12 0.0296 0.0142 0.001 MISE(R(t), ^ R MVU ) 0.02 0.0048 0.0038 0.0022 MISE(R(t), ^ R Bayes (t)) 0.0075 0.0022 0.001 0.0003 MISE(R(t), ^ R EB (t)) 0.01 0.0046 0.0043 0.0019 MISE(R(t), ^ R KD (t)) 0.194 0.132 0.099 0.0896 Table3.4:MISEfortheReliabilityEstimates 55

PAGE 68

n, =1 20 50 100 200 MISE(h(t), ^h ML ) 0.0043 0.0007 0.0006 0.0001 MISE(h(t), ^h MVU ) 0.00584 0.0013 0.0004 0.0003 MISE(h(t), ^h Bayes (t)) 0.0014 0.0001 0.00001 0.0000009 MISE(h(t), ^h EB (t)) 0.004 0.0008 0.00009 0.000007 MISE(h(t), ^h KD (t)) 0.05 0.003 0.001 0.00009 n, =2 20 50 100 200 MISE(h(t), ^h ML ) 0.022 0.0055 0.001 0.0001 MISE(h(t), ^h MVU ) 0.0046 0.0041 0.0009 0.00003 MISE(h(t), ^h Bayes (t)) 0.0025 0.00015 0.0001 0.000003 MISE(h(t), ^h EB (t)) 0.0039 0.0022 0.0009 0.000009 MISE(h(t), ^h KD (t)) 0.09 0.078 0.055 0.001 n, =4 20 50 100 200 MISE(h(t), ^h ML ) 0.024 0.011 0.008 0.0001 MISE(h(t), ^h MVU ) 0.011 0.009 0.0004 0.0001 MISE(h(t), ^h Bayes (t)) 0.0033 0.0001 0.00004 0.00001 MISE(h(t), ^h EB (t)) 0.009 0.0009 0.0001 0.00003 MISE(h(t), ^h KD (t)) 0.1 0.07 0.06 0.06 Table3.5:MISEfortheFailureRateFunctionEstimates 56

PAGE 69

n, =1 20 50 100 200 MISE(H(t), ^ H ML ) 0.133 0.032 0.021 0.0035 MISE(H(t), ^ H MVU ) 0.222 0.044 0.019 0.001 MISE(H(t), ^ H Bayes (t)) 0.057 0.034 0.002 0.00019 MISE(H(t), ^ H EB (t)) 0.25 0.03 0.019 0.00025 MISE(H(t), ^ H KD (t)) 0.32 0.099 0.058 0.0059 n, =2 20 50 100 200 MISE(H(t), ^ H ML ) 0.45 0.23 0.09 0.055 MISE(H(t), ^ H MVU ) 0.25 0.11 0.087 0.0018 MISE(H(t), ^ H Bayes (t)) 0.061 0.041 0.008 0.0006 MISE(H(t), ^ H EB (t)) 0.18 0.08 0.055 0.001 MISE(H(t), ^ H KD (t)) 0.5 0.4 0.16 0.1 n, =4 20 50 100 200 MISE(H(t), ^ H ML ) 0.44 0.27 0.15 0.09 MISE(H(t), ^ H MVU ) 0.29 0.11 0.099 0.033 MISE(H(t), ^ H Bayes (t)) 0.12 0.087 0.055 0.009 MISE(H(t), ^ H EB (t)) 0.22 0.10 0.084 0.018 MISE(H(t), ^ H KD (t)) 0.64 0.53 0.44 0.18 Table3.6:MISEfortheCumulativeFailureFunctionEstimat es 57

PAGE 70

n, =1 20 50 100 200 MISE( t c ^ t ML ) 0.244 0.048 0.012 0.0023 MISE( t c ^ t MVU ) 0.16 0.055 0.008 0.0009 MISE( t c ^ t Bayes ) 0.093 0.034 0.0001 0.00001 MISE( t c ^ t EB ) 0.10 0.045 0.006 0.0002 MISE( t c ^ t KD ) 0.26 0.12 0.033 0.009 n, =2 20 50 100 200 MISE( t c ^ t ML ) 0.25 0.16 0.09 0.0088 MISE( t c ^ t MVU ) 0.16 0.099 0.0001 0.00002 MISE( t c ^ t Bayes ) 0.099 0.054 0.0001 0.00006 MISE( t c ^ t EB ) 0.12 0.091 0.004 0.0003 MISE( t c ^ t KD ) 0.37 0.30 0.21 0.08 n, =4 20 50 100 200 MISE( t c ^ t ML ) 0.227 0.176 0.153 0.0023 MISE( t c ^ t MVU ) 0.134 0.099 0.09 0.00001 MISE( t c ^ t Bayes ) 0.11 0.083 0.003 0.00001 MISE( t c ^ t EB ) 0.123 0.097 0.006 0.00002 MISE( t c ^ t KD ) 0.36 0.261 0.21 0.15 Table3.7:MISEfortheTargetTime t c 58

PAGE 71

3.7Conclusion Inthischapterweanalyticallyderivedandnumericallystu diedtheeectivnessof themaximumlikelihood,minimumvarianceunbiased,Bayes, empiricalBayes,and kerneldensityestimatesofReliabilityforamodiedGumbe lfailuremodel.Kernel densitymethodologywasappliedundertheassumptionthatt hemodiedGumbel probabilitydistributiondidnottthefailuredata.Based onourworkweconclude thefollowing: (i)Asexpected,thenumericalsimulationindicatesthatth eBayesestimateunder thenaturalconjugatepriorandempiricalBayesestimatear eclosertothetrue estimatesofreliabilitythantheirordinarycounterparts .Theincreaseinprior variancehadnoeectontherankingofourestimates. (ii)Theve-stepprocedureintroducedinchapter2produce dclosekerneldensity estimates.Thekerneldensityestimatesofreliabilitysim ulatedfromthesubject failuremodelandusingLiu-Tsokosoptimalbandwidthgivea sgoodestimatesas theparametricmodelswithoutanyanalyticassumptions.It isaneectiveand relativelysimplealternativetoparametricestimation.H owever,theincreasein posteriorvarianceslightlydecreasedtheclosenessofour estimatebasedonthe valueofMISE. (iii)Themosteectiveestimatesofreliability,failurer ate,cumulativefailurerate, andtargettimeweretheBayesandempiricalBayes,followed bytheMVU,ML, andkerneldensityandestimates.Theincreaseinsamplesiz egaveuscloser estimates,butdidnotchangetherankingoftheestimateee ctiveness. 59

PAGE 72

4SensitivityBehaviorofBayesianReliabilityfortheGumb el FailureModelforDifferentPriors 4.1Introduction Inchapter3wederivedtheBayesianreliabilityestimatesf orthemodiedGumbelfailuremodelunderthenaturalconjugatepriorandsquarederro rlossfunction.Tsokos (1999)analyzedthemodiedextremevaluedistributionand derivedtheBayesian estimatesofthereliabilityfunctionundertheinruenceof thesquarederrorlossfunctionandthreedierentpriors:generaluniform,exponenti al,andinvertedGamma priordensityfunctions.However,nosensitivityanalysis underthethreepriorswas performed.Ouraimistoinvestigatetherobustnessofrelia bilityforthemodied GumbelfailuremodelintheBayesiansettingsubjecttothei nverseGaussian,invertedgamma,Gamma,generaluniform,diuse,andnon-para metrickerneldensity prior.Whenanengineerorascientistperformsreliability analysisofagivensystem andisunabletoidentifyawelldenedprobabilitydistribu tionfunctionfortheprior density,thenon-parametrickerneldensityestimateofthe priorprovidesforagood alternative.Weintroducethekerneldensityestimateasap riortocharacterizethe behaviorofthemodiedGumbellocationparameterandascer tainitseectiveness whencomparedtootherveparametricpriors.Wewillshowth atthenon-parametric kerneldensitypriorperformsaswellastheparametricprio rs.Themaindierence betweenthenon-parametrickerneldensityandparametricp riorsisthatthekernel densitypriorisdistributionfreeandmostrexibleinmodel ingtheprobabilisticstructureofpriorinformation.Inadditiontotheanalyticframe work,wehaveperformed anextensivenumericalanalysistocomparetheBayesanalyt icreliabilityestimates 60

PAGE 73

underthesubjectpriors.Afterderivingtheexpressionfor eachreliabilityestimate, wewillperformanumericalstudytoestablishitseectiven ess.Ageneralcomparison betweenthereliabilityestimatesunderthekerneldensity andallotherpriorswillbe conductedinthenumericalanalysissection.Wewillshowth atthekerneldensity priorperformsaswellastheparametricones.Thischapteri ssetoutasfollows.In section4.2wedescribethepriorsunderinvestigation.Sec tion4.3presentsthereliabilityestimatesundereachpriorandnumericalstudycompa ringeachwiththetrue reliabilityfunction.Thepairwisecomparisonoftheeect ivenessforeachparametric priorandthekerneldensitypriorispresentedinsection4. 4.Conclusionsaregiven insection4.5. 4.2ThePriors Theprobabilitydensityandreliabilityfunctionsofthemo diedGumbeldistribution underconsiderationare: f ( t ; )= [ g ( t )] ( 1) [ g ( t )] 0 ;> 0 ;t> 0(4.2.1) and R ( t ; )=1 [ g ( t )] ; (4.2.2) where g ( t ; )= e e t ; ( ; )= e Thesensitivityanalysiswewillengageinwillusesixdier entpriorscharacterizing respectivelytheprobabilisticbehavioroftheparameter inthefailuremodel.They aretheinvertedgamma,theinverseGaussian,Gamma,thegen eraluniform,the diuse,andourkerneldensityprior.Toourknowledge,thep erformanceofthenonparametrickerneldensityestimatesofthepriordistribut ionhasnotbeenperformed andisintroducedhereforthersttime.Wehavealreadyshow ninchapter3that theGammaprobabilitydistributionisthenaturalconjugat epriorforparameter 61

PAGE 74

.TheBayesianreliabilityestimateswillbederivedusingt hesquarederrorloss functionbecauseofitsanalytictractability.Giventruer eliabilityfunctionR(t)and itsestimate ^ R ( t ),thesquarederrorlossfunctionforreliabilityisdened as L SE ( ^ R n ( t ) ;R ( t ))=( ^ R n ( t ) R ( t )) 2 : Thepriorsarerespectivelygivenbelow: (i)TheinverseGaussianpriorwithparameters and isgivenby g 1 ( ; ; )=( 2 3 ) 1 2 e ( ) 2 2 2 ;> 0 ;;> 0 : (4.2.3) (ii)Theinvertedgammaprobabilitydistributionisdened asfollows: g 2 ( ; ; )= 1 ( ) ( ) +1 e ;> 0 ;;> 0 : (4.2.4) (iii)TheGammaprobabilitydistributionisdenedas g 3 ( ; ; )= ( ) 1 e : (4.2.5) (iv)Thegeneraluniformprobabilitydistributionisgiven by g 4 ( ; ;;b )= ( b 1)( ) b 1 b ( b 1 b 1 ) ; 0 : (4.2.6) Thediusedistributionisobtainedfromthegeneralunifor mdistributionby settingb=0andletting 0and !1 (v)Thekerneldensitypriorforsamplesizenandbandwidthh isgivenby g 5 ( ; n;h )= 1 nh n X i =1 K ( i h ) : (4.2.7) 62

PAGE 75

ForthekernelfunctionweshallusetheGaussiankernel,giv enby K ( u )= 1 p 2 e 0 : 5 u 2 ; 1
PAGE 76

(iii)Forcomparisonpurposeswecalculatethemeanintegra tedsquareerror(MISE) betweenthetruemodiedGumbelreliabilityandthecorresp ondingBayescounterpartsundervedierentpriors.Wecalculatetheestima tedMISEacrossall samplesnwhere MISE ( R ( t ) ; ^ R n ( t ))= N X i =1 Z ( ^ R n ( t ) R ( t )) 2 dt N acrossallfailuretimest,forthereliabilityfunctionR(t ),reliabilityfunction estimate ^ R n ( t ),andNthenumberofsimulationsperformed. Aschematicdiagramofthecompletestepbystepprocessofth enumericalanalysis ispresentedinFigure4.1. 64

PAGE 77

Figure4.1:NumericalStudyofPriors 65

PAGE 78

4.3MainResults Let t 1 ,..., t n denotelifetimesofnsystemsfromapopulationwhoselifeti mesare subjecttoalifetestterminatedafternsystemshavefailed .Thejointprobabilityof observingnindependentfailuresattimes t 1 ,..., t n isgivenby f ( t j )= L ( t ; )= n n Y i =1 g 0 ( t i )[ g ( t i )] 1 : (4.3.1) Uponsimplication,weobtain f ( t j )= L ( t ; )= n n Y i =1 g 0 ( t i ) g ( t i ) n e G ; (4.3.2) where G = n X i =1 lng ( t i ) ReliabilityUndertheInverseGaussianPrior TheposteriorprobabilitydistributionfortheinverseGau ssianpriorisgivenby h IGS ( j t )= L ( t ; ) g 1 ( ; ; ) Z 1 0 L ( t ; ) g 1 ( ; ; ) d h IGS ( j t )= ( 2 3 ) 1 2 n e G ( ) 2 2 2 0 : 798(2 G 2 + ) 0 : 25 0 : 5 n 0 : 5+ n 0 : 25+0 : 5 n e ( B K ( n 0 : 5 ; p 2 G 2 + p )) 1 h IGS ( j t )=0 : 5 1 : 5+ n e 0 : 5 2 2 G 2 + 2 + 2 2 (2 G 2 + ) 0 : 25+0 : 5 n 0 : 5 n 0 : 25 0 : 5 n ( B K ( n 0 : 5 ; p 2 G 2 + p )) 1 (4.3.3) BK(m,n)representsthesecondorderBesselfunctionthatsa tisesthedierential equation n 2 y 00 + ny 0 ( n 2 + m 2 ) y =0 ;y 0 66

PAGE 79

TheBayesianreliabilityestimatecorrespondingtotheinv erseGaussianpriorwith thesquareerrorlossfunctionisgivenby ^ R IGS ( t )=1 Z 1 0 g ( t ) h IGS ( j t ) d ^ R IGS ( t )=1 (2 G 2 + 2 lng ( t ) 2 ) 0 : 25 0 : 5 n (2 G 2 + ) 0 : 25+0 : 5 n B K ( t ; n;; )(4.3.4) where BK ( t ; n;; )= B K ( n 0 : 5 ; p 2 G 2 + 2 lng ( t ) 2 p ) B K ( n 0 : 5 ; p 2 G 2 + p ) Table4.1listsMISEbetweenthetruereliabilityandBayesi anreliabilityestimate undertheinverseGaussianpriorandsquareerrorlossfunct ion,forn=30,50,and 100. Error ^ R IGS =1 ^ R IGS =2 ^ R IGS =4 MISE(n=30) 0.0022 0.012 0.01 MISE(n=50) 0.0002 0.009 0.005 MISE(n=100) 0.0002 0.001 0.001 Table4.1:MISEUnderInverseGaussianPrior ReliabilityUndertheInvertedGammaPrior Similarly,fortheinvertedgammapriortheposteriordensi tyisgivenby h IGM ( j t )= 0 : 5 n ( ) +1 e G n +1 ( G ) n 2 BK ( n ; 2 p G ) ; (4.3.5) andtheBayesianreliabilityestimateby ^ R IGM ( t )=1 G 0 : 5 n 0 : 5 ( G lng ( t )) 0 : 5 n +0 : 5 BK ( n ; 2 p ( G lng ( t ))) BK ( n ; 2 p G ) : (4.3.6) 67

PAGE 80

Table4.2listsMISEbetweenthetruereliabilityandBayesi anreliabilityestimate undertheinvertedgammapriorandsquareerrorlossfunctio n. Error ^ R IGM =1 ^ R IGM =2 ^ R IGM =4 MISE(n=30) 0.0023 0.014 0.012 MISE(n=50) 0.0002 0.01 0.008 MISE(n=100) 0.0002 0.01 0.002 Table4.2:MISEUnderInvertedGammaPrior ReliabilityUndertheGammaPrior FortheGammaprior,theposteriordensitygivenby h GM ( j t )= ( + G ) n + ( n + ) n + 1 e ( + G ) ; (4.3.7) theBayesianreliabilityestimateis ^ R GM ( t )=1 ( + G + G lng ( t ) ) n + : (4.3.8) TheGammafamilyofpriordensitiesof isalsothenaturalconjugatefamilyforthe modiedGumbeldistribution.Table4.3listsMISEbetweent hetruereliabilityand BayesianreliabilityestimateundertheGammapriorandsqu areerrorlossfunction. Error ^ R GM =1 ^ R GM =2 ^ R GM =4 MISE(n=30) 0.0021 0.017 0.08 MISE(n=50) 0.0002 0.02 0.07 MISE(n=100) 0.0002 0.014 0.004 Table4.3:MISEUnderGammaPrior 68

PAGE 81

ReliabilityUndertheGeneralUniformandDiusePriors Forthegeneraluniformpriortheposteriordensityisgiven by h GU ( j t )= e G b n ( Z e G n b d ) 1 ; (4.3.9) andtheBayesianreliabilityestimateby ^ R GU ( t )=1 Z g ( t ) e G n b d Z e G n b d : (4.3.10) ThegeneraluniformPDFrestrictsthedomainoftheparamete r toaninterval [ ].Ifwelackknowledgetodene and wemayletb=0and[ ] [0, 1 )inthegeneraluniformdensity.Parameter thenhasadiuseprioroverthe nonnegativerealline,andtheBayesianestimateofreliabi litybecomes ^ R D ( t )=1 ( G G lng ( t ) ) n +1 (4.3.11) Tables4.4and4.5listMISEbetweenthetruereliabilityand Bayesianreliabilityestimateunderthegeneraluniformprioranddiusepriorandsqu areerrorlossfunction, respectivelyConsideringthepriorswehaveintroducedsof arandMISEvaluesin Error ^ R GU =1 ^ R GU =2 ^ R GU =4 MISE(n=30) 0.004 0.02 0.04 MISE(n=50) 0.005 0.032 0.0031 MISE(n=100) 0.0002 0.06 0.0026 Table4.4:MISEUnderGeneralUniformPrior tables4.1-4.5,asthesamplesizeincreases,thevalueofMI SEdecreases.Thenatural conjugateGammapriordoesnotalwaysproducetheclosestes timates.Theincrease invarianceinGumbelvariance doesseemtoslightlyincreasethevalueofMISE. 69

PAGE 82

Error ^ R D =1 ^ R D =2 ^ R D =4 MISE(n=30) 0.212 0.12 0.12 MISE(n=50) 0.005 0.1 0.11 MISE(n=100) 0.003 0.09 0.086 Table4.5:MISEUnderDiusePrior ReliabilityUnderKernelDensityPrior Forthekerneldensityprior,theposteriordistributionis givenby h K ( j t )= L ( t ; ) 1 nh n X i =1 K ( i h ) Z 1 0 L ( t ; ) 1 nh n X i =1 K ( i h ) d (4.3.12) andtheBayesianreliabilityestimateby ^ R K ( t )=1 Z 1 0 g ( t ) h K ( j t ) d: (4.3.13) TheBayesreliabilityestimategivenby4.3.13doesnothave ananalyticformandmust beevaluatednumerically,undertheoptimalbandwidthsele ctionmethod(Silverman (1986)).Thekerneldensitypriorisempiricalinnatureand soitdoesnotassume anyparticulardistributionfortheparameter .Inthenextsection,foreachof thesimulatedsamplesunderparametricpriors,wewillcalc ulateBayesestimatesof reliabilityunderthekerneldensityprior.Wewillperform apairwisecomparisonof thereliabilityestimatesunderthekerneldensityandpara metricpriorsbycomparing thevaluesofMISE. PropertiesofReliability HereweshowthattheBayesianreliabilityestimatessatisf ythebasicpropertiesof thereliabilityfunction lim t !1 ^ R ( t )=0 70

PAGE 83

and lim t 0 ^ R ( t )=1 : IfBK(m,n)representsthesecondorderBesselfunctionthat satisesthedierential equation n 2 y 00 + ny 0 ( n 2 + m 2 ) y =0 ;y 0 thenitsatisesthefollowingproperties: lim n !1 BK ( m;n )=0(4.3.14) and lim n 0 BK ( m;n )= 1 (4.3.15) Sincealso lim t !1 lnt =0 lim t 0 lnt = 1 itisapparentthatallofourBayesianreliabilityestimate ssatisfythetwobasicasymptoticpropertiesofreliability.Inthenextsectionwepres entanumericalstudyto pairwisecomparetheeectivenessofthevedierentprior sandthekerneldensity priorinestimatingthereliabilityfunction. 4.4NumericalComparisonofPriors Inthissectionwepresentpairwisecomparisonbetweenther eliabilityestimatesunderthekerneldensityandothervepriordistributionfunc tionsused.Theresultsaresummarizedintables4.6-4.8.Forsamplesizen(n= 30,50,100),and =1(smallvariance), =2(mediumvariance)and =4(largevariance), eachtablepresentsthemeanintegratedsquareerrorbetwee ntheparametricestimateandtruereliabilitypairedwiththecorrespondingi ntegratedmeansquare valuebetweenthenon-parametrickerneldensityestimatea ndtruereliability.Note that ^ R IGS ^ R IGM ^ R GM ^ R GU ^ R D ^ R K correspondtotheBayesianreliabilityesti71

PAGE 84

mateswhenthepriorsarerespectivelytheinverseGaussian ,theinvertedgamma, Gamma,thegeneraluniform,thediuse,andthekerneldensi typriordensityfunctions.Aswecanseefromeachtable,theincreaseinsamplesi zenproducesestimates thatareclosertothetrueGumbelreliability,irrespectiv eofthesizeof .Thekernel densitypriorproducedcloserestimatesthanthediuseand generaluniformpriors. Itisclearthatforallntestedkerneldensitypriorgivesus goodresultswithoutany assumptions.However,theincreasein decreaseditseectiveness.Theassumed naturalconjugatepriordoesnotalwaysproducetheclosest estimates. Error ^ R IGS ^ R K ^ R IGM ^ R K ^ R GM ^ R K ^ R GU ^ R K ^ R D ^ R K MISE(n=30) 0.0022,0.02 0.0023,0.03 0.0021,0.06 0.004,0.06 0.212,0.08 MISE(n=50) 0.0002,0.008 0.0002,0.009 0.0002,0.009 0.005,0.04 0.005,0.004 MISE(n=100) 0.0002,0.001 0.0002,0.001 0.0002,0.005 0.0002,0.01 0.003,0.004 Table4.6:AverageIntegratedMeanSquareErrorsfor =1 Error ^ R IGS ^ R K ^ R IGM ^ R K ^ R GM ^ R K ^ R GU ^ R K ^ R D ^ R K MISE(n=30) 0.012,0.15 0.014,0.017 0.017,0.029 0.02,0.02 0.12,0.1 MISE(n=50) 0.009,0.01 0.01,0.011 0.02,0.025 0.032,0.03 0.1,0.07 MISE(n=100) 0.001,0.01 0.01,0.01 0.014,0.02 0.06,0.06 0.09,0.08 Table4.7:AverageIntegratedMeanSquareErrorsfor =2 Error ^ R IGS ^ R K ^ R IGM ^ R K ^ R GM ^ R K ^ R GU ^ R K ^ R D ^ R K MISE(n=30) 0.01,0.01 0.012,0.09 0.08,0.08 0.04,0.06 0.12,0.06 MISE(n=50) 0.005,0.05 0.008,0.01 0.07,0.06 0.0031,0.07 0.11,0.01 MISE(n=100) 0.001,0.03 0.002,0.07 0.004,0.003 0.0026,0.009 0.086,0.09 Table4.8:AverageIntegratedMeanSquareErrorsfor =4 72

PAGE 85

4.5Conclusion InthischapterweobtainedBayesianreliabilityestimates withthesquareerrorloss andthemodiedGumbelfailuremodel,whoseparameterischa racterizedbythe inverseGaussian,invertedgamma,Gamma,generaluniform, anddiusepriors.Additionally,weassumeapriorstructurebasedonkerneldens ityestimation.Wealso performedanextensivenumericalsimulation.Basedonoura nalyticaldevelopments andcomputersimulation,weconcludethefollowing: (i)Bayesianreliabilityestimatesaresensitivetothecho iceofthepriordistribution.Thenaturalconjugateprior,theGammaprobabilitydi stribution,does notalwaysleadtotheclosestestimatesofreliability. (ii)TheinverseGaussian,invertedgammaandGammapriorsp roducealmostidenticalestimatesofBayesianreliabilityandareasymptotic allyecient.Thegeneraluniformanddiusepriorsproducereliabilityestimat esthatarenotasclose tothetrueReliability,buttheydoprovidemorerexibility ifweareuncertain aboutthepriorchoice. (iii)Thekerneldensitypriorperformedverywellwhencomp aredwithitsparametric counterparts.Itgenerallyperformedbetterthanthegener aluniformanddiuse priorsforn=30andn=50andthuswerecommendthekerneldens itypriorbe usedoverthegeneraluniformanddiusepriors.Themeanint egratedsquare error(MISE)correspondingtoitwaseitherbetterforsmall samples(n=30)or approximatelythesameastheinverseGaussian,invertedGa mmaandGamma priors.WenoticedaslightincreaseinMISEasthevalueof wasincreased from1to2to4. 73

PAGE 86

5BayesianModelingofTargetTimefortheGumbelFailure Model:RandomLocationandScaleParameters 5.1Introduction TheobjectofthepresentstudyistousetheclassicalGumbel ordoubleexponential probabilitydistributiontocharacterizethefailuretime sofagivensystem,bothinthe ordinaryandBayesiansettings.IntheBayesiansetting,we assumethattheprior probabilitydensityfunctionistheJerey'snon-informat ivepriorunderthemean squareerrorlossfunction.Weareinterestedinobtainingo rdinaryandBayesian estimatesofatargettime t ,subjecttoadesiredandspeciedreliability.That is,foragivensystemwhatisthetimetofailure, t ,withatleast(1)100% assurance.Forexample,wewanttobeatleast95%certaintha tthesystemwillbe operabletotime t 0 : 05 .WedevelopbothordinaryandBayesianestimatesof t and introduceLindley'sapproximationprocedurethatisusedt oobtainnumericalresults thatillustratetheusefulnessofthestudy. 5.2TheGumbelModel FortheGumbelmodel,theprobabilitydistributionfunctio n(PDF)andthecumulativedistributionfunction(CDF)ofthefailuretimeTaregi ven,respectively,by f ( t )= 1 e t e t ; 1 0(5.2.1) 74

PAGE 87

and F ( t )= exp f exp f ( ( t ) gg (5.2.2) where and arethelocationandscaleparameters,respectively.Thism odelhas beenusedinreprotection,insuranceproblems,predictio nofearthquakemagnitudes, carbondioxidelevelsintheatmosphere,highreturnlevels ofwindspeedsinthedesign ofstructuresamongothers.Inthepresentstudy,weshallap plythesubjectmodel inreliabilityanalysisand,morespecically,Bayesianre liabilitymodeling. 5.3ReliabilityModeling Let t 1 t 2 t 3 ,..., t n bethefailuretimesthatfollowtheGumbelPDFgivenby(5.2. 1). ThelikelihoodfunctionL( ),isgivenby L ( ; )= n exp f n X i =1 t i n X i =1 exp ( t i ) g anditslogarithmicformis LogL = nln n X i =1 ( t i ) n X i =1 e ( t i ) (5.3.1) Themaximumlikelihoodestimates(MLE)for and canbeobtainedfromthe likelihoodfunctionsbysolvingthefollowingequations ^ + t i e t i = ^ e t i = ^ = t (5.3.2) and ^ = ^ ln f 1 n e t i = ^ g (5.3.3) Equations(5.3.2)and(5.3.3)arenotanalyticallytractab leandmustbesolvednumericallytoobtainapproximateMLE'sof and ,thatis^ and^ .Bytakingthe naturallogarithmofbothsidesofequation(5.2.2)andsolv ingfortweobtainthe 75

PAGE 88

expressionforthetargettime t underthedesiredreliability1givenby t = ( ln ( ln ( ))(5.3.4) Thus,bytheinvariancepropertyoftheMLE'swecanobtainth eMLEofthetarget time t ^ t =^ ^ ( ln ( ln ( ))(5.3.5) Classicalestimatesandcondenceintervalsfor t canbeobtainedusingthemethod ofmaximumlikelihoodandthenormalapproximationfordie rentextremevalue models.Inthepresentstudy,weshallexaminetheestimatio nof t foranextreme valuemodelinaBayesiansettingunderaspeciedpriorandm eansquareerrorloss function. 5.4BayesianApproachtotheGumbelModel IntheBayesianapproachweregard and behavingasrandomvariableswith ajointPDF ( ; ).Weshallinvestigatethepointestimatorof t forJerey's non-informativeprior.Jerey'sNon-informativePriorJerey'snon-informativepriorchoosestheprior ( ; )tobeproportionalto p detI ( ), whereI( )istheexpectedFisherinformationmatrix.Thatis, I (^ ; ^ )= E @ 2 @ 2 ln L @ 2 ln L @@ 2 @ 2 ln L @@ 2 @ 2 @ 2 ln L : UsinglogLasgivenin(5.3.1),weobtainI( )as I ( )= n 2 n 2 (1 r ) n 2 (1 r ) n 2 f r 2 2 r +2+ (2 ; 2) g : (5.4.1) 76

PAGE 89

where r istheEuler'sconstantand ( p;q )= 1 ( p ) 1 Z 0 t p 1 e qt 1 e t dt: Hence, det ( I ( ))= K 4 implyingthattheJerey'snon-informativepriorisgivenb y ( ; )= 1 2 (5.4.2) Weremarkthat isanimproperpriorPDF.Consistentwiththeaimoftheprese nt studyinidentifyingthetargettime t ,weproceedtoobtainitsanalyticalform. PosteriorDistributionTheposteriorprobabilitydensityfunctionof( ; )giventhefailuretimes t 1 ;:::;t n isgivenby ( ; j t 1 ;t 2 ;:::;t n )= L ( ; j t 1 ;:::;t n ) ( ; ) R 1 0 R 1 1 L ( ; j t 1 ;t 2 ;:::;t n ) ( ; ) dd where L ( ; j t 1 ;:::;t n )isgivenby(5.3.1).Weshallrstcomputethemarginal probabilitydensityfunction,thatis, Z 1 0 Z 1 1 L ( ; j t 1 ;:::;t n ) ( ; ) dd: Usingtheprior ( ; )= 1 2 > 0andletting x = n P i =1 t i ,weobtain Z 1 0 Z 1 1 L ( ; j t 1 ;:::;t n ) ( ; ) dd 77

PAGE 90

= Z 1 0 n 2 Z + 1 1 e x= e ( n= ) e e ( = ) P ni =1 e t i = dd: (5.4.3) Let u = e ,and a = P nl =1 e ( t 1 = ) ,theexpression(5.4.3)canbewrittenas Z 1 0 Z 1 1 L ( ; j t 1 ;:::;t n ) ( ; ) dd (5.4.4) = 1 Z 0 n 1 e x= Z 1 0 u n 1 e au dud =( n ) Z 1 0 n 1 e x= a n d =( n ) Z 1 0 v n 1 e xv ( n X l =1 e t i v ) n dv =( n ) Z 1 0 v n 1 ( n X l =1 e v ( t i + t ) ) n dv where x = n P l =1 t i = n t BayesianEstimationof t forJerey'sPrior TheBayesestimateof t = ln( ln( )) forsquarederrorlossisgivenby ^t B = E ( t j t 1 ;t 2 ;:::;t n )= Z 1 0 Z 1 1 [ ln( ln( )] L ( ; j t 1 ;t 2 ;:::;t n ) ( ; ) dd or ^ t B = R 1 0 R 1 1 [ ln( ln )] L ( ; j t 1 ;t 2 ;:::;t n ) ( ; ) dd ( n ) R 1 0 v n 1 ( P e ( t i + t ) v ) n dv (5.4.5) 78

PAGE 91

Proceedingaswedidbeforeforobtainingthemarginalproba bilitydistributionwe canwrite E ( j ~ t )= Z 1 0 v n 2 e xv Z 1 0 (ln u ) u n 1 e au dudv (5.4.6) where a = n X l =1 e t i v ; and E ( j ~ t )=( n ) Z 1 0 v n 2 ( n X l =1 e v ( t i + t ) ) n dv: (5.4.7) Hence, E ( ^ t j t 1 ;:::;t n )= R 1 0 v n 2 e xv R 1 0 (ln u ) u n 1 e au dudv ( n ) R 1 0 v n 1 [ P nl =1 e v ( t i + t ) ] n dv +( ln( ln )) R 1 0 v n 2 [ P e v ( t i + t ) ] n dv R 1 0 v n 1 [ P nl =1 e v ( t i + t ) ] n dv where a = n X l =1 e t i v and t = n P l =1 t i n : ToevaluatetheaboveexpressiontoobtainapproximateBaye sianestimatesof t ,we shalluseLindley'sapproximationmethod.TheLindleyApproximationLet I = R u ( ) v ( ) e L ( ) d R v ( ) e L ( ) d where =( 1 ; 2 ;:::; k ),avectorofparameters.Also,let L =Log(likelihoodfunction) 79

PAGE 92

Notethat I istheposteriorexpectationof u ( ~ )giventhefailuredata,foraprior v ( ).Denoteby u 1 = @u @ 1 u 2 = @u @ 2 u 11 = @ 2 u @ 2 1 u 22 = @ 2 u @ 2 2 p = ( 1 ; 2 ) p 1 = @p @ 1 ; p 2 = @p @ 2 L 20 = @ 2 L @ 2 1 ; L 02 = @ 2 L @ 2 2 L 30 = @ 3 L @ 3 1 ; L 03 = @ 3 L @ 3 2 and 11 =( L 20 ) 1 and 22 =( L 02 ) 1 Furthermore, E ( u ( ) j ~ t )= u ( ^ 1 ; ^ 2 )+ 1 2 ( u 11 11 + u 22 22 )+ P 1 u 1 11 + P 2 u 2 22 + 1 2 ( L 30 u 1 2 11 + L 03 u 2 2 22 + L 21 u 2 11 22 + L 12 u 1 22 11 ) evaluatedat( ^ 1 ; ^ 2 ),where ^ 1 and ^ 2 aretheMLEsof 1 and 2 .Thetargettime fortheGumbelmodelgivenby t B = b = u ( ; ) ; where 1 = and 2 = .Also, u 1 =1and u 2 = b where b =ln( ln )where u 11 =0and u 22 =0.Thus,wecanwrite P ( 1 ; 2 )= ( ; )= 1 2 80

PAGE 93

and P 1 =0 andP 2 = 2 3 : Let^ and^ betheclassicalMLEsfor and ,respectively.Furthermore,wehave L = n exp ( n X i =1 ( t i ) n X i =1 exp( t i ) ) or ln L = n ln n X i =1 ( t i ) n X i =1 e t i : Thus, @ ln L @ = n 1 n X i =1 e t i and L 2 ; 0 = @ 2 ln L @ 2 = 1 2 n X i =1 e t i : Also, @ ln L @ = n + n X i =1 ( t i ) 2 1 2 n X i =1 e ( t i ) ( t i ) andL 0 ; 2 = @ 2 ln L @ 2 = n 2 2[ n X i =1 ( t i )] 1 3 + 2 3 n X i =1 e t i ( t i ) 1 4 n X i =1 e ( t i ) ( t i ) 2 whichcanbeexpressedas n 2 ( n X i =1 ( t i )) 1 3 +( n X l =1 e t i ) ( t i ) 3 1 4 n X i =1 e t i ( t i ) 2 or n 2 [ n X l =1 2( t i )[1 e ( t i ) ]] 1 3 +( n X i =1 e ( t i ) ( t i ) 2 ) 1 4 : 81

PAGE 94

Weproceedtond L 3 ; 0 and L 0 ; 3 ,thatis L 3 ; 0 = @ 3 ln L @ 3 = 1 3 n X i =1 e t i and L 0 ; 3 = @ 3 ln L @ 3 = 2 n 3 +6[ n X i =1 ( t i )[1 e ( t i ) ]] 1 4 +6[ n X i =1 e ( t i ) ( t i ) 2 ] 1 5 1[ n X i =1 e ( t i ) ( t i ) 3 ] 1 6 : Also, L 21 = @ @ ( @ 2 ln L @ 2 )=[ n X i =1 e ( t i ) ] 2 3 [ n X i =1 e ( t i )( t i ) ] 1 4 and L 12 = @ @ ( @ 2 ln L @ 2 ) or L 12 = 2 3 ( n n X i =1 e t i )+ 4 4 X ( t i ) e t i 1 5 n X i =1 ( t i ) 2 e t i Thus,aBayesianapproximateestimatefor t isgivenby ^ t B = ^ t ( MLE )+ P 2 u 2 22 + 1 2 ( L 30 2 11 + L 03 u 2 2 22 + L 21 u 2 11 22 + L 12 22 11 )(5.4.8) evaluatedattheMLEof and ,^ and^ 5.5NumericalAnalysis Inthissectionwepresentanumericalstudyinordertocompa rethemaximumlikelihoodandBayesestimatesfordeterminingthetargettimeo ftheGumbelfailure modelsubjecttospeciedreliability.Ournumericalsimul ationwasconductedinthe followingmanner: (i)Undertheassumptionthatthelocationparameter andthescaleparameter 82

PAGE 95

behaverandomlyandindependently,wesimulatedm(m=50,10 0,200) locationparametersfromthenormaldistribution.Inorder tostudytheeects ofthepriorvarianceonourestimates,wesimulatedlocatio nparametersfromthe normaldistributionwithmean25andvariancesequalto1,4, and9respectively. (ii)Weassumedthescaleparameterfollowstheuniformdist ribution.However,in ordertoseewhateectstheincreaseofvariancehasonoures timates,welet equalto1,2and4respectively. (iii)Usingtheobtainedmpairsof and ,wegeneratedn(n=50,100,200) observationsfromtheGumbelPDFandcalculatedboththemax imumlikelihood andBayesestimatesofthetargettime. (iv)Forcomparisonpurposes,wecalculatedtheabsoluteva lueofthedierence betweenthetruetargettimeandthecorrespondingMLandBay esestimates for99%reliability. Aschematicdiagramofthecompletestep-by-stepprocessof thenumericalanalysis ispresentedingure5.1.Duetothesizeofoursimulationsomeofthenumericalresult saregivenintables 5.1-5.3under99%reliability.Ineachtablewepresentthes izeofthepriorsamplem usedtocalculatetheBayesestimate B ,while^ and^ aretheMLestimatesofthe locationandscaleparameters. j t ^t j and j t ^ t B j representtheabsolutevalue ofthedierencebetweenthetruetargettime,andmaximumli kelihoodandBayes targettimeestimatesrespectively.Aswecanseefromtable 5.1,bykeepingtheprior samplesizem=50andpriorvariancexedandvaryingthesamp lesizeofthefailure modelfromn=50ton=200,theabsolutevalueofthedierence j t ^t j and j t ^ t B j decreases.Thisbehaviorisconsistentasweincrease andn,exceptthat wenoticeasignicantimprovementintheMLestimate.Intab le5.2andtable5.3 weincreasethepriorsamplesizemto100and200andpriorvar ianceto4and9 83

PAGE 96

Figure5.1:NumericalStudyoftheGumbelFailureTime m n B ; ^ B ; ^ j t ^t j j t ^t B j 50 50 25.0526,25.2053 1,0.9683 0.2011 0.0938 50 100 25.0526,25.1542 1,0.8594 0.3165 0.2548 50 200 25.0526,24.999 1,0.9608 0.008 0.01 50 50 25.0526,25.2433 2,1.9397 0.2827 0.1485 50 100 25.0526,25.1688 2,1.9276 0.2267 0.1607 50 200 25.0526,25.1454 2,1.9591 0.1552 0.1214 50 50 25.0526,25.455 4,4.047 0.3312 0.0362 50 100 25.0526,25.0701 4,4.043 0.2167 0.0805 50 200 25.0526,25.0342 4,3,951 0.1543 0.0726 Table5.1:ComparisonBetweenMLandBayesianEstimatesofR eliabilityTime: N(25,1), =1,2,4, =0.01 84

PAGE 97

m n B ; ^ B ; ^ j t ^ t j j t ^ t B j 100 50 25.0701,25.1942 1,0.9796 0.1552 0.0611 100 100 25.0701,25.1968 1,0.9696 0.1416 0.1226 100 200 25.0701,25.1252 1,0.9584 0.1185 0.0921 100 50 25.0701,25.145 2,1.771 0.4792 0.3587 100 100 25.0701,25.145 2,1.8397 0.3195 0.2566 100 200 25.0701,25.06 2,1.774 0.2195 0.1956 100 50 25.0701,25.42 4,4.05 0.2167 0.1805 100 100 25.0701,24.92 4,3.904 0.07 0.069 100 200 25.0701,24.77 4,3.85 0.01 0.01 Table5.2:ComparisonBetweenMLandBayesianEstimatesofR eliabilityTime: N(25,2), =1,2,4, =0.01 m n B ; ^ B ; ^ j t ^t j j t ^t B j 200 50 21.982,21.964 1,0.8696 0.1805 0.0921 200 100 21.982,22.01 1,0.9562 0.0904 0.0372 200 200 21.982,21.03 1,0.999 0.0472 0.0214 200 50 21.982,21.991 2,1.8243 0.278 0.148 200 100 21.982,22.011 2,2.151 0.203 0.21 200 200 21.982,21.845 2,2.1965 0.1303 0.130 200 50 21.982,22.1561 4,3.7306 0.2834 0.291 200 100 21.982,22.201 4,3.8402 0.1625 0.159 200 200 21.982,21.756 4,3.6615 0.091 0.101 Table5.3:ComparisonBetweenMLandBayesianEstimatesofR eliabilityTime: N(25,3), =1,2,4, =0.01 respectivelyandalsoobservethattheabsolutevalueofthe dierence j t ^ t j and j t ^ t B j decreases.Theincreaseinthepriorvariancehasnoeecton thebehavior ofourestimates.Thisisconsistentasweincrease andn,andweagainnoticea signicantimprovementintheMLestimate.Inalmosteveryc asetheBayesestimate 85

PAGE 98

isclosertothetruetargettimethanitsmaximumlikelihood counterpart. 5.6Conclusion Asexpected,theMonteCarlosimulationindicatesthattheB ayesestimateunderthe non-informativepriorisclosertothetruereliabilitytim ethanitsmaximumlikelihood counterpart.However,thefollowingndingsareinorder: (i)Anincreaseinthepriorsamplesizeforthelocationpara meterhasnoeecton thebehavioroftheestimates. (ii)AnincreaseinthesamplesizeofthesimulatedGumbelda taresultsinthe improvementofboththemaximumlikelihoodandBayesianest imates. (iii)Whenweincreasethevarianceofthepriordistributio nfrom1to4to9andthe varianceofthesimulatedGumbeldatafrom1to2to4,wenotic easignicant improvementinthemaximumlikelihoodestimate.Wetherefo reconcludethat forlargesamplesizeandhighvariancethereisverylittled ierencebetween themaximumlikelihoodandtheBayesestimatesofthetarget timesubjectto speciedreliability. 86

PAGE 99

6TheChoiceoftheLossFunctionUnderBayesianParameter Estimation 6.1Introduction OneofthecentralissuesinBayesiananalysisisthechoiceo fthelossfunction.Inthe Bayesianframework,thelossfunctionmeasurestheconsequ enceofadecisiondierent fromtheonethatwouldyieldthebestpossibleresult.Theob jectiveindecision problemsistochoosetheactionthatminimizestheexpected valueofthelossfunction withrespecttotheposteriordistribution.Extensiveamou ntofresearchhasledto theformulationofseveralcriteriaforthechoiceofonelos sfunctionoveranother(see Berger,1985;Makov,1994).Theaimofthepresentchapteris tointroducethenonparametrickerneldensitymethodasasimpleandeectivecr iterion,andcompareits performancewiththreeothercommonlyusedcriteria,namel ytheMinimaxcriterion, theMakovcriterion,andthegoodnessoftcriterion.Wewil luseresultsfromchapter 2andthevestepproceduretoderivethebestkerneldensity estimate,andshow thatthekerneldensityestimateisuniforminpickingtheco rrectlossfunction.After introducingthefourcriteria,wewillderivetheBayesiane stimatefortheGumbel failuremodel,withthelocationparameterbehavingasaran domvariableundera familyoflossfunctionsandnaturalconjugateprior.Wewil lusetheconceptof eciencywhencomparingBayesianestimates.Inthenumeric alstudysectionwewill showthatourkerneldensitycriterionissuperiortotheoth erthreecriteria. 87

PAGE 100

6.2LossFunctionSelectionCriteria Let L = f L 0 ;L 2 g beafamilyoflossfunctionsand ^ B 0 and ^ B 2 theBayesian estimatesoftheparameter obtainedusingthelossfunctions L 0 and L 2 .The followingfourcriteriamaybeusedtotestwhichestimateof theparameter is better:Criterion1:MinimaxCriterionUsingPosteriorRisksTheBayesianestimate ^ B 0 is"better"thantheBayesianestimate ^ B 2 if max ( L 0 ( ^ B 0 ) ; L 2 ( ^ B 0 )) 1 : (6.2.1) NotethattheBayesianestimates ^ B 0 and ^ B 2 areequallyecientiftheeciency Eff MM isapproximatelyequaltoone. Criterion2:Makov'sCriterionTheBayesianestimate ^ B 0 is"better"thantheBayesianestimate ^ B 2 if sup j @ @t L 0 ( ; ^ B 0 ) j = inf L i L sup j @ @t inf ^ L ( ; ^ ) j : (6.2.2) 88

PAGE 101

Criterion3:GoodnessofFitCriterionGivenanunderlyingfailurePDF f ( t j ),theBayesianestimate ^ B 0 is"better"than theBayesianestimate ^ B 2 if f ( t j ^ B 0 )isabetterttothegivendatathan f ( t j ^ B 2 ). ThiscriterioncanbeexpresseddierentlyusingtheKolmog orov-Smirnovstatistic forthegoodnessofttest.Let t 1 ... t n beasamplefromacumulativedistribution functionF,andlet F beacorrespondingempiricalprobabilitydistributionfun ction. Thestatistic D = sup j F ( t ) F ( t ) j iscalledthetwosidedKolmogorov-Smirnovstatistic.Thus ^ B 0 is"better"than ^ B 2 iftheKolmogorov-SmirnovStatistic D ^ B 0 correspondingtothecumulativedistribution F ( t j ^ B 0 )issmallerthan D ^ B 2 correspondingtothecumulativedistribution function F ( t j ^ B 2 ). Criterion4:ProbabilityDensityCriterionGivenanunderlyingcumulativedistributionfunction F ( t ; )anditsnon-parametric kerneldensityestimate ^ F ( t ; ),theBayesianestimate ^ B 0 is"better"thanthe Bayesianestimate ^ B 2 ifthedensityestimate ^ F ( t ; ^ B 0 )producesasmallerintegratedmeansquareerrorwhencomparedwith ^ F ( t ; )thanthekerneldensityestimate ^ F ( t ; ^ B 2 ).Theintegratedsquareerror(ISE)ofanestimate ^ F ( t )isdened as ISE ( ^ F ( t ) ;F ( t ))= Z ( ^ F ( t ) F ( t )) 2 dt (6.2.3) Thecriterioncanbestatedintermsofrelativeeciency.Th erelativeeciencyof ^ F ( t ; ^ B 0 )withrespecttoanother ^ F ( t ; ^ B 2 )wedeneastheratiooftheISEofthe estimate ^ F ( t ; ^ B 0 )tothatof ^ F ( t ; ^ B 2 ),thatis Eff Density = ISE ( ^ F ( t ; ^ B 2 ) ; ^ F ( t ; )) ISE ( ^ F ( t ; ^ B 0 ) ; ^ F ( t ; )) (6.2.4) 89

PAGE 102

TheBayesianestimate ^ B 0 willbelessormoreecientthan ^ B 02 iftherelative eciencyissmallerorgreaterthanone,respectively.Ifth erelativeeciencyis approximatelyone,thenweshallconsiderthemequallyeci ent. NextweproceedtoapplythefourcriteriatoaGumbelfailure model,under theassumptionthatthelocationparameterbehavesasarand omvariableunderthe naturalconjugatepriorandafamilyoflossfunctions. 6.3MainResults Inordertoimplementandcomparetheabovecriteria,thefol lowingfamilyofloss functionshallbeconsidered: L i ( ; ^ )= ( ^ ) 2 i ;i =0 ; 2 : (6.3.1) ThefailuremodelunderconsiderationisthemodiedGumbel model.Givenfailure times t 1 ,..., t n theprobabilitydensityfunctionisgivenby f ( t ; )= [ g ( t )] ( 1) [ g ( t )] 0 ;> 0 ; where g ( t ; )= e e t ; ( ; )= e and and arethelocationandscaleparametersrespectively.TheLik elihood functionisgivenby f ( t j )= L ( t ; )= e G n E ( t i ) ; where E ( t i )= g 0 ( t i )[ g ( t i )] 1 andG = n X i =1 ln ( g ( t i )) Weassumethattheparameter behavesasarandomvariablethatisprobabilisticallycharacterizedbyitsnaturalconjugateprior,name lytheGammaprobability 90

PAGE 103

distributiongivenby g ( ; ; )= ( ) 1 e ;> 0 : (6.3.2) Therefore,theposteriordensityfunctionunderthenatura lconjugateprioris f ( j t )= ( + G ) n + ( n + ) n + 1 e ( + G ) : (6.3.3) TheBayesianestimateof correspondingtothesquarederrorlossfunction L 0 is givenby ^ B 0 = Z 1 0 f ( j t ) d: ^ B 0 = Z 1 0 ( + G ) n + ( n + ) n + 1 e ( + G ) d = ( n + +1) ( + G )( n + ) ^ B 0 = n + + G (6.3.4) TheBayesianestimateof correspondingtothelossfunction L 2 isgivenby ^ B 2 = R 1 0 f ( j t ) d R 1 0 f ( j t ) 2 d or ^ B 2 = Z 1 0 ( + G ) n + ( n + ) n + 2 e ( + G ) d Z 1 0 ( + G ) n + ( n + ) n + 3 e ( + G ) d = ( + G )( n + 1)( n + ) ( + G ) 2 ( n + )( n + 2) ; whichproduces ^ B 2 = n + 2 + G (6.3.5) 91

PAGE 104

AstheStrongLawoflargenumbersgives lim n !1 n G = ;a:s: because n G isminimumvarianceunbiasedestimatorof ,theBayesianestimatesof theparameter givenby(6.3.4)and(6.3.5)convergealmostsurelyto as n !1 Criterion1:MinimaxCriterionUsingPosteriorRisksTheposteriorriskof L 0 evaluatedat ^ B 2 isgivenby: L 0 ( ; ^ B 2 )= Z 1 0 ( ^ B 2 ) 2 f ( j t ) d L 0 ( ; ^ B 2 )= Z 1 0 ( ^ B 2 ) 2 ( + G ) n + ( n + ) n + 1 e ( + G ) d L 0 ( ; ^ B 2 )= (2+ n + )+( 2 n 2 +4)(1+ n + )+( n + 2) 2 ( n + ) ( + G ) 2 ( n + ) ; whichsimpliesto L 0 ( ; ^ B 2 )= ( n + +4) ( + G ) 2 : (6.3.6) Similarly,theposteriorriskof L 2 evaluatedat ^ B 0 isgivenby L 2 ( ; ^ B 0 )= Z 1 0 ( ^ B 0 ) 2 f ( j t ) d L 2 ( ; ^ B 0 )= ( n + ) ( n + ) + ( 2 n 2 )( n + 1) ( n + ) + ( n + ) 2 ( n + 2) ( n + ) ; whichgives 92

PAGE 105

L 2 ( ; ^ B 0 )= n + +2 ( n + 1)( n + 2) : (6.3.7) Equations(6.3.6)and(6.3.7)yieldthefollowingexpressi onforeciency: Eff MM = L 0 ( ; ^ B 2 ) L 2 ( ; ^ B 0 ) : Eff MM = ( n + +4)( n + 2)( n + 1) ( + G ) 2 (2+ n + ) (6.3.8) Criterion2:Makov'sCriterionEquations(6.3.4)and(6.3.5)canbecombinedintothefollo wingformoftheBayesian estimateof ^ Bi = n + i + G ;i =0 ; 2 : (6.3.9) Thusweget L i ( ; ^ Bi )= Z 1 0 ( ^ Bi i ) 2 ( + G ) n + ( n + ) n + 1 e ( + G ) d; or L i ( ; ^ Bi )= ( + G ) i 2 (1 i + n + ) ( n + ) : (6.3.10) Fork=1,...,n,weobtain @ @t k L i ( ; ^ Bi )= ( i 2)( + G ) i 3 (1 i + n + ) ( n + ) @ @t k G (6.3.11) Equation(6.3.11)showsthatMakov'scriterionissatised fori=2,whichimplies that L 2 willbechosenover L 0 .Accordingtothiscriterion,theBayesianestimate ^ B 2 is"better"than ^ B 0 ,irrespectiveofthesamplesizen. Thefollowingtwocriteriarequirenotheoreticalcalculat ionsandimplementvery simpleprocedurestocomparetheeectivenessofthetwoBay esianparameterestimates B 0 and B 2 93

PAGE 106

Criterion3:GoodnessofFitCriterionAKolmogorov-Smirnovgoodnessofttestisconductedtocom pareprobabilitydensityfunctions f ( t j ^ B 0 )and f ( t j ^ B 2 )andascertainwhichprovidesabettertfor theobservations t i .Aschematicdiagramofthecompletestepbystepprocessund er thiscriterionispresentedinFigure6.1.Criterion4:ProbabilityDensityCriterionThekerneldensityestimate ^ F ( t )ofthecumulativedistributionfunction F ( t )is constructedusingthefourstepprocedurefromchapterone. UsingtheBayesian parameterestimates B 0 and B 2 ,parametricestimatesofthecumulativedensityfunctions ^ F ( t ; ^ B 0 )and ^ F ( t ; ^ B 2 )areconstructed.Integratedsquareerrors ISE( ^ F ( t ), ^ F ( t ; ^ B 0 ))andISE( ^ F ( t ), ^ F ( t ; ^ B 0 ))arecalculatedtoinordertocalculate theeciencyestimate.Aschematicdiagramofthecompletes tepbystepprocess underthiscriterionpresentedinFigure6.2. Inthefollowingsectionwepresentanumericalstudyinorde rtoascertainthe eectivenessofeachofthecriteriainchoosingthecorrect lossfunction. 94

PAGE 107

Figure6.1:GoodnessofFitCriterionImplementationChart 6.4CriteriaComparison Ournumericalsimulationwasconductedinthefollowingman ner: (i)Inordertostudytheeectsofthesizeofpriorparameter s and ,underthe assumptionthatparameter behavesrandomlyundertheGammaprior,we 95

PAGE 108

Figure6.2:KernelDensityCriterionImplementationChart generatedm=30parameters fromtheGammadistributionforeachofthe pairsofpriorparametersgivenintable6.1. (ii)Usingtheobtainedmvaluesofparameter ,n(n=20,50,100,200)samples fromtheGumbeldistributionweregeneratedundertheassum ptionthat = 1. (iii)UndereachofthesamplestheBayesianestimates ^ B 0 and ^ B 0 oftheparameter 96

PAGE 109

^ ^ 0.8 1.2 4.3 6.1 8 3.2 0.5 2.5 25 20 2 2.0001 6.3 12.1 30 0.5 Table6.1:PriorParameterValues werecalculatedcorrespondingtothelossfunctions L 0 and L 2 (iv)Makov'scriterionchoosesthelossfunction L 2 over L 0 ,andsoconsiders ^ B 2 tobeabetterestimateof over ^ B 0 regardlessofsamplesizen.TheMinimax Eciency Eff MM ,theKolmogorov-Smirnovstatistics D 0 and D 2 correspondingtothecumulativedistributions F ( t j ^ B 0 )and F ( t j ^ B 2 ),andtheDensity Eciency Eff Density werecalculatedtocomparethecriteriaabove. Table6.2presentsthepercentagecorrespondingtothenumb eroftimesacriterion issuccessfulinpickingthelossfunctionthatyieldsthebe stBayesianestimateofthe parameter .ItindicatesthattheKernelDensitycriterionisthemosts uccessful, followedbythegoodnessoft,Makov'sandMinimaxcriteria .Makov'scriteriondoes notdependonthesamplesizen. MinimaxPost.RiskCrit. Makov'sCrit. GoodnessofFitCrit. DensityCrit. 39% 42% 78% 88% Table6.2:PercentageofSuccessoftheDierentCriteria Duetothesizeofournumericalsimulation,someofourresul tsarepresentedin table6.3.Thetablecontainsthevaluesofthepriorparamet ers and ,Bayesian 97

PAGE 110

estimates ^ B 0 and ^ B 2 correspondingtothelossfunctions L 2 and L 2 ,KolmogorovSmirnovstatistics D 0 and D 2 ,andMinimaxandDensityeciencyestimates Eff MM and Eff Density .Aswecanseefromtable6.3,assamplesizenincreases,thee ciency oftheMinimaxcriteriondoesnotalwaysuniformlyapproach one. 98

PAGE 111

, n ^ B 0 ^ B 2 Eff MM D 0 D 2 Eff Density 0.8,1.2 20 0.869 0.786 0.7068 0.1752 0.1844 0.6787 50 0.408 0.3921 0.1627 0.101 0.0872 1.4304 100 0.4741 0.4647 0.2224 0.0734 0.0691 1.8931 200 0.6106 0.6046 0.3710 0.0411 0.0375 0.903 4.3,6.1 20 0.8428 0.773 0.6725 0.4612 0.4314 0.7074 50 0.4197 0.404 0.1724 0.1583 0.1661 1.471 100 0.4795 0.4703 0.2275 0.0745 0.0813 2.136 200 0.6122 0.606 0.3792 0.069 0.0656 0.906 8,3.2 20 1.079 1.003 1.1134 0.3993 0.3733 0.1 50 0.4585 0.4428 0.2062 0.1215 0.1325 1.262 100 0.5032 0.4939 0.2507 0.0903 0.0851 1.360 200 0.6288 0.6227 0.3934 0.1025 0.099 0.8763 0.5,2.5 20 0.9086 0.8199 0.7716 0.1418 0.1715 4.266 50 0.8732 0.8386 0.7452 0.1349 0.1434 1.682 100 1.0412 1.0204 1.0724 0.0717 0.0791 0.2523 200 0.9699 0.9602 0.9358 0.0415 0.0452 1.764 25,20 20 1.123 10.733 1.23 0.2008 0.194 0.371 50 0.9956 0.9691 0.9767 0.1056 0.1156 51.36 100 1.0962 1.0787 1.19 0.0667 0.623 0.6802 200 1.0035 0.995 1.003 0.0413 0.038 2.48 2,2.0001 20 0.9971 0.9064 0.9347 0.0931 0.0826 1176.35 50 0.907 0.8722 0.804 0.1362 0.1234 1.9625 100 1.0622 1.0414 0.1165 0.0992 0.0946 0.452 200 0.9795 0.9698 0.9545 0.0934 0.0959 2.193 6.3,12.1 20 1.7305 1.599 2.849 0.5796 0.5973 1.071 50 2.930 2.82 8.413 0.3478 0.3602 1.062 100 4.691 4.603 21.78 0.3117 0.318 1.069 99

PAGE 112

200 5.396 5.344 28.97 0.1557 0.1592 1.049 6.3,12.1 20 2.043 1.887 3.9709 0.7829 0.7946 1.032 50 3.85 3.71 14.498 0.5914 0.6009 1.029 100 6.172 6.05 37.703 0.4221 0.4288 1.025 200 9.356 9.266 87.099 0.2956 0.2989 1.023 30,0.5 20 39.219 37.65 1502.8 0.1813 0.1687 0.889 50 26.34 25.688 684.644 0.2218 0.2171 0.826 100 23.114 22.758 529.9 0.1039 0.0984 0.798 200 22.01 21.82 482.3 0.0646 0.0625 0.826 Table6.3:NumericalComparisonoftheDierentCriteriaUs edfortheChoiceoftheLoss Function 6.5Conclusion Inthischapter,wecomparedseveralcriteriathatareusedi ntheselectionprocessof thelossfunctionleadingtothebestBayesianestimateofth eGumbelFailuremodel parameter denedinChapter3.OurMonteCarlosimulationleadstothef ollowing conclusion: (i)Thefourcriteriausedinselectingthelossfunctiontha tgivesthebestBayesian estimateoftheparameter arenotallequivalent.TheMinimaxcriterionusing posteriorrisksandtheMakov'scriterionareapproximatel yequivalent.Amajor drawbackoftheMakov'scriterionisthatitpicksthebestes timateirrespective ofthesamplesizen. (ii)Theeectivenessofthefourcriteriaremainsconstant regardlessofthevalueof thepriorparametersorsamplesize. (iii)Amongallfourcriteria,themostconsistentistheKer nelDensitycriterion.Itis computationallyintensive,easytoapplyinpracticeandre quiresnoanalytical calculations.Theconceptofeciency,alongwiththenotio nofintegratedmean squareerrorgivesveryconsistentresultswithrespecttot hechoiceoftheloss 100

PAGE 113

functionleadingtothebestBayesianestimateoftheparame ter .Thekernel densitycriterionistheonlyforwhichtheeciencyuniform lyapproachesone asthesamplesizeisincreased. (iv)AnadvantageoftheKernelDensitycriterionisthatitt ellsushowclosethe Bayesianestimateistothetruestateofnature.Theclosert heintegratedsquare erroristozero,themorelikelythecorrespondingBayesian estimateiscloserto thetruestateofnature. 101

PAGE 114

7KernelDensityEstimationasanAlternativetotheGumbel DistributioninModelingQuantilesandReturnPeriodsforFloo d Prevention 7.1Introduction Extremevalueprobabilitydistributionshavebeeneectiv elyappliedinthemodeling ofrooddata.SomeimportantreferencesareBenson(1968),K irby(1969),North (1980),Katzetal.(2002),MorrisonandSmith(2001),North rop(2004),andYue (2005).Aroodisusuallycausedbyariverthathasoverrowed itsbanksduring periodsofhighrunoandmaybepredictedbystudyingtheann ualmaximumriver streamrow.Amongthemostcommonprobabilitydistribution usedinmodelinghydrologicalextremesistheGumbelprobabilitydistributio n.However,severalrecent studieshaveshownthatroodsseemtohaveheaviertailsthan aGumbelprobabilitydistribution,whichmayyieldthesmallestquantilespo ssible(seeFarquharsonet al.(1992),Turcotte(1994),TurcotteandMalamud(2003)). Otherstudies(Wilks (1993)andKoutsoyiannisandBaloutsos(2000))haveextend edthescepticismforthe Gumbelprobabilitydistributionbyshowingthatitunderes timatesthelargestrainfall amounts.Inthepresentstudy,theproblemofestimationofr oodquantilesandreturn levelsisstudiedinparametricandnon-parametricsetting s.Weprovideanextensive analysisoftheannualmaximumriverstreamrowoftheHillsb oroughRiver,Florida, datingfrom1940to2006,byapplyingtheanalyticndingsan dmethodologyfrom thepreviouschapters.Themotivationbehindourstudyisth ataroodcausedby theHillsboroughRiverwouldhaveamajorimpactontheecono myandpopulation oftheTampaBayarea.Usingthreegoodnessofttests,namel ytheKolmogorov102

PAGE 115

Smirnov,PearsonChi-square,andAnderson-Darling,weint endtoshowthatthe Gumbelprobabilitydistributionprovidesagoodoverallt totheHillsboroughRiver annualmaxima,butfailstoprovideagoodtinthetails.Usi ngtheresultsfromthe previouschapters,wewillshowthatthenon-parametricker neldensityestimation proceduresolvestheproblembyprovidingcloserestimates inthetails.Parametrically,ourestimationwillbeperformedinordinaryandBa yesiansettings.Some Bayesiananalysishasbeenappliedtorooddataestimation( seeFortinetal.(1997) andVanNortwijk(2001)).Thepresentstudyisorganizedasf ollows.Insection7.2, weintroducetheannualmaximastreamrowdataset,andusing thefrequencyfactor analysis,showthatthemaximumstreamrowmaybemodeledusi ngtheGumbel probabilitydistribution.Insection7.3,wepresentsixdi erentmodelsthatcanbe usedtoestimatethequantileandreturnperiodfunctions.M odel1estimatestheparametersoftheGumbelprobabilitydistributionusingthem aximumlikelihood(ML) method.Wetestwhethermaximumannualstreamrowexhibitst rendswithrespect totimeundermodels2and3.Jackknifeisacomputationallyi ntensiveprocedure thathasbeenshowntoreducestandarderrorofparameterest imates,sofollowingthe workofPfanzagelandWefelmeyer(1978)andHahn,Kuerstein erandNewey(2002), weusethejackknifeprocedureundermodel4toreducethesta ndarderrorofML parameters.WeconducttheBayesiananalysisinmodel5byas sumingthatboth thelocationandscaleparametersbehaverandomlyunderJe rey'snon-informative priorandsquareerrorlossfunction.Lastly,undermodel6w eassumethatnoneof theparametricmodelsthedata,andusethevestepprocedur efromchapter2to ndtheoptimalnon-parametrickerneldensityquantileand returnperiodestimates. Weshowthatalthoughnon-parametricmodelisinferiorover alltotheparametric models,itprovidesabettertintherighttailoftheannual maximastreamrow, andisthereforemoreaccurateinpredictingreturnperiods ofroods.Insection7.4, wecomparethesixmodelsandpresentourrecommendations.C oncludingremarks aregiveninsection7.5. 103

PAGE 116

7.2PreliminaryExplorationoftheExtremeStreamFlowData Floodsoccurwhenthewaterheight,commonlymeasuredbyast reamgauge,passes somepredeterminedlevel,whichisusuallytakenasthebank -fullstage.Whenthe streamchannelcannolongeraccommodatetheincreaseddisc harge,itoverrowsits banks.Discharge,orstreamrowisthevolumeofwaterthatpa ssesaspeciclocation inagivenperiodoftime,andismeasuredbycombiningmeasur ementsofariver's waterrowvelocityandcross-sectionalarea.Theunitsused arecubicfeetpersecond (cfs).Floodsarerelatedtoextremestreamrowandhaveamaj orimpactonthelocal populationandeconomy.Inthisstudy,weareinterestedinm odelingtheextreme rowoftheHillsboroughRiverbyanalyzingitsannualpeakdi schargeincubicfeet persecond(cfs)nearZephyrhills,HillsboroughCounty,Fl orida.Inourstudy,the fourdierentroodstagesandthecorrespondingstreamrows aregivenbelow: Actionstage=1507cfs Floodstage=2431cfs Moderateroodstage=4982cfs Majorroodstage=6329cfs Thepeakannualstreamrowdataareavailableforyears19402006intable7.7and wereprovidedbytheU.S.GeologicalSurvey.Thetableprese ntstheexactdateon whichthepeakannualstreamrowwasrecorded,alongwiththe waterlevelgauge heightandmaximumstreamrow.Asonecanseefromthetable,b etween1940and 2006therehavebeenveannualpeakeventscorrespondingto roodsatthemoderate levelandtwoatthemajorlevel.Figure7.1showshowtheannu almaximumstream rowhasvariedfrom1940to2006.InordertoascertainhowwelltheGumbelprobabilitydistri butiontsthedata,we employthemethodoffrequencyfactoranalysis(Castilloan dHadi(2005)).This methodisanexploratorytoolthatprovidesaprobabilitypl otoftheobservedversus expectedvaluesforaparticularprobabilitydistribution .Ifthedatafollowaparticular 104

PAGE 117

1940195019601970198019902000 020004000600080001000012000 YearAnnual Peak Streamflow Figure7.1:AnnualMaximaStreamFlowfortheHillsboroughR iver1940-2006 probabilitydistribution,wewouldexpectthedatapointst ofollowastraightline. ThefrequencyfactoranalysisfortheGumbelprobabilitydi stributionconsistsofthe followingsteps: (i)Ranktheobserveddatai=1,...,nindescendingorder. (ii)Assignprobabilitiesp(plottingpositions)toeachda tapoint T p using p i = n +12 i 0 : 44 (iii)Calculatetheexpectedstreamrowvaluesusing SF estim = SF mean + K T SF stdev where 105

PAGE 118

SF mean =MeanpeakannualStreamrow SF stdev =StandarddeviationofthepeakannualStreamrow K T =Frequencyfactordenedas K T = p 6 (0 : 5772+ ln ( ln ( T p T p 1 ))) (iv)Plottheobservedpeakstreamrowandtheexpectedstrea mrowvaluesasa functionof K T (v)Forndegreesoffreedomplottheupperandlowercondenc ebandsgivenby SF estim t r 1+1 : 14 K T +1 : 1 K 2 T n SF stdev Usingthevestepsoutlinedabove,weobtainedthefrequenc yfactorplotandthe 95%condencebandsdisplayedbygure7.2,wheretheobserv edvaluesareshowed ascircles.Aswecansee,theyfollowalineartrend,sowesus pecttheGumbel probabilitydistributiontobeagoodt.However,wealsono ticesomediscrepancies intheloweranduppertails,especiallytheextremevalueof annualstreamrowof 12600,whichwasrecordedin1960.Next,weproceedtomodelquantileandreturnperiodfunctio nsfortheHillsborough Riverannualmaximumstreamrow.Wedosoparametricallybya pplyingtheGumbel probabilitydistribution,andnon-parametricallyusingk erneldensityestimation.We intendtoshowthatthekerneldensitymethodprovidesclose rextremerighttail estimatesofquantilesandreturnperiodsthantheparametr icmodels. 7.3PeakStreamFlowQuantileandReturnPeriodModeling Let x 1 ;:::;x n denotetheannualpeakstreamrowfornyearsfromagivenloca tion. Inmodelingextremeeventsinhydrology,twofunctionsareo fparticularinterest:the 106

PAGE 119

-10123 0200040006000800010000 KTObserved Figure7.2:Ninety-vePercentCondenceBandFrequencyFa ctorPlotfortheAnnual PeakStreamFlowquantilefunctionqin(0,1),denedas F 1 ( q )= inf ( xR : q F ( x )) ; (7.3.1) whereF(x)isthecumulativedistributionfunction,andthe returnperioddenedas RP ( x )= 1 1 F ( x ) : (7.3.2) IfXistheannualmaximumpeakstreamrow,thenthequantilef unctionQ(F)isthe valueweexpectXtoexceedwithprobability(1-F)duringthe yearofinterest.That is,thereis(1-F)%chancethatXwillexceedQ(F).Thereturn period,alsocalledthe recurrenceinterval,isdenedastheaverageperiodofretu rnoftheextremeevent.In otherwords,itisthereciprocaloftheprobabilityofexcee danceinoneyear, 1 1 F ( x ) 107

PAGE 120

ItistheexpectedtimebetweenexcedancesofsizeQ(F).Peak streamrowexceedance hasbeenestimatedusingtwoapproaches,thefrequentistan dnon-frequentist.The conventionalfrequentistapproachassumesthatthestream rowfollowsaparticular distributionandestimatestheparametersofthatdistribu tionusingthemethodof maximumlikelihood.Theaimofthenon-frequentistestimat ionisthatthereisno "correct"functionalformthatgeneratesrandomoutcomesa nditisnon-parametric innature.Theprobabilitydensity,cumulativedensity,qu antile,andreturnperiod functionsfortheGumbelprobabilitydistributionmodelar erespectivelygivenby: f ( x )= 1 e x e x ; 1 0(7.3.3) F ( x )= e e x (7.3.4) Q ( F )= ln ( ln ( F ))(7.3.5) and RP ( x )= 1 1 e e x (7.3.6) where and arethelocationandscaleparameters,respectively. Next,weproceedtodevelopsixcandidatemodels,inorderto providethebestt totheannualpeakstreamrowdata.Inassessinghowwellthem odeltthedata, weshallusethreegoodnessofttests.Thepopular 2 goodnessofttestisanonparametrictestthatdividestheabscissaoftheCDFandcalc ulatestheprobability p i foreachoftheintervals.Thenumberofobservationsexpect edtofallinthe i th intervalisn p i and n i istheobservednumber.ThefollowingPearson 2 statisticis usedtocalculatethep-value 2 = n X i =1 ( np i n i ) 2 np i TheKolmogorov-Smirnov(KS)testusesasastatisticthelar gestabsolutedierence betweentheempiricaldistributionfunction ^ F e ( x ),andthespeciedtheoreticalCDF 108

PAGE 121

F(x).Thestatisticusedtocalculatethep-valueisdistrib utionfreeandisgivenby D = sup x j ^ F e ( x ) F ( x ) j TheAnderson-DarlingtestisamodicationoftheKolmogoro v-Smirnovtestthat givesmoreweighttothetails.However,unliketheKStestwh ichisdistributionfree, itmakesuseofthespecicdistributionbeingtestedandall owsforamoresensitive test.Thep-valueisobtainedusingtheteststatistic A 2 = n 1 n n X i =1 (2 i 1)( ln ( F ( x i ))+ ln (1 F ( x n i +1 ))) whereF(x)istheCDFofthespecieddistributionand x i aretheordereddata.This testisusefulindetectingtheextentoflackoftinthetail sofaproposedmodel. Theproposedsixdierentcandidatemodels,whichmaybeuse dinmodelingthe maximumannualstreamrowareasfollows:Model1:TheMaximumLikelihoodModelTherstmodelweconsiderisdevelopedbyestimatingparame ters and using themaximumlikelihoodmethod.Assumingindependenceofth edata,thelikelihood L ( ; )istheproductofthedensitiesofequation7.3.3fortheobs ervations x 1 ;:::;x n Mathematically, L ( ; )= nln n X i =1 x i n X i =1 e x i (7.3.7) Themaximumlikelihoodestimatesof and ,^ and^ ,aretakentobethosevalues whichmaximizethelikelihood L ( ; )andarecalculatedbysolvingthefollowing equations ^ + x i e x i = ^ e x i = ^ = x 109

PAGE 122

and ^ = ^ ln f 1 n e x i = ^ g Theaboveequationsarenotanalyticallytractableandmust besolvednumericallyto obtainapproximateMLE'sof and .Thecorresponding(1)100%condence intervalsfor and forlargesamplesizenaregivenby ^ z 1 = 2 p Var (^ )(7.3.8) ^ z 1 = 2 p Var (^ )(7.3.9) where Var (^ )= 2 ( r 2 2 r +2+ (2 ; 2)) n ((1+ (2 ; 2)) and Var (^ )= 2 n (1+ (2 ; 2)) Then,MLestimateofthequantilefunction, ^ Q ,andthecorresponding(1)100% condenceintervalaregivenby ^ Q =^ ^ ln ( ln ( F ))(7.3.10) and ^ Q z 2 q Var ( ^ Q )(7.3.11) for Var ( ^ Q )= ^ 2 ( r 2 2 r +2+ (2 ; 2)) n ((1+ (2 ; 2)) +( ln ( ln ( F ))) 2 ^ 2 n (1+ (2 ; 2)) 2 ln ( ln ( F )) ^ 2 (1 r ) n (1+ (2 ; 2)) where r =0.57722istheEuler'sconstantand ( p;q )= 1 ( p ) 1 Z 0 t p 1 e qt 1 e t dt: 110

PAGE 123

TheMLestimateofthereturnperiodfunctionisgivenby: ^ RP = 1 1 e e x ^ ^ (7.3.12) Table7.1givesthemaximumlikelihoodestimatesoftheloca tionandshapeparameters,alongwiththecorresponding95%condenceinterval sandthegoodnessoft p-values.Thep-valuesindicatethattheMLmodelprovidesa verygoodt.However,thesignicantdropinthep-valueunderthetailsensi tiveAnderson-Darlingtest suggeststhegoodnessoftmaynotbesogreatinbothleftand righttails.Figures 7.3and7.4presentthequantileandreturnperiodfunctions underthemethodof maximumlikelihood. ^ ,95%CI ^ ,95%CI K-S 2 A-D 1764.454,(1428.3,2101.5) 1314.909,(1065.7,1564.2) 0.4212 0.4339 0 : 1
PAGE 124

12000 10000 60 8000 5040 6000 3020 4000 10 0 2000 0 RP Under ML Figure7.4:StreamFlowReturnPeriodFunctionUnderML Models2and3:TimeDependentLocationParameterFigure7.1suggeststhatextremestreamrowcouldpossiblye xhibittrendswithrespecttotime.Toinvestigatethis,thefollowingvariation softheMLmodelwerealso tted: Model2: ( t )= a + b ( Year t 0 +1) ; = constant (7.3.13) athreeparametermodelwith allowedtovarylinearlywithrespecttotime; Model3: ( t )= a + b ( Year t 0 +1)+ c ( Year t 0 +1) 2 ; = constant (7.3.14) afourparametermodelwith allowedtovaryquadraticallywithrespecttotime (where t 0 denotestheyeartherecordsstarted).Higherorderpolynom ialsareoften betteratdescribingadataset,buttheirprojectionsintot hefuturetendtoructuate toowildly,andinthecaseofvariability,theyshrinkorexp andtooquickly.Thiswas oftenfoundtobethecasewithttingcubicorhigherorderpo lynomials;thusthose 112

PAGE 125

modelswerenotconsidered.Thestandardlikelihoodratiot estwasusedtodetermine whetherthetrendsdescribedmymodels2and3weresignican t.SincetheMLmodel isasubmodelofbothequations(7.3.13)and(7.3.14),astan dardwayofdetermining thebesttmodelisthelikelihoodratiotest.If L 2 isthemaximumlikelihoodforthe threeparametermodel1and L 1 isthemaximumlikelihoodforthetwoparameter model,thenunderthesimplermodeltheteststatistic =-2log( L 1 / L 2 )wouldbe assumedtobedistributedasaChi-squarevariablewith1deg reeoffreedom(since thenumberofparametersdierby1).Inhypothesistestingt hatproblemsthiswould beasymptoticallytrueasthenumberofdatatendstoinnity .Thus,at95%level ofsignicance,thesimplerparametermodelwouldbeprefer edif-2log( L 1 / L 2 ) < 21 ; 0 : 95 =3.841.Inpractice,becauseofthelackofcompleteindepen denceofthe annualmaxima,thiswouldprobablyhavetobeinterpretedco nservatively.Usingthe likelihoodratiotest,wetestedtoseewhethermodelstwoan dthreeprovidedbetter tthantheMLestimates.Table7.2summarizestheparameter estimates,along withthevaluesofloglikelihood.Aswecanseefromthetable ,since-2log( L 1 / L 2 ) =1146.77-1145.21=1.56 < 3.841= 21 ; 0 : 95 ,itfollowsbythestandardlikelihood ratiotestthatmodel1shouldbepreferredandweseenoevide nceoflineartrend inthelocationparameterwithrespecttotime.Since-2log( L 1 / L 3 )=1146.771145.07=1.7 < 5.991= 22 ; 0 : 95 itfollowsbythestandardlikelihoodratiotestthat model1shouldbepreferredandweseenoevidenceofquadrati ctrendinthelocation parameterwithrespecttotime.Sincethereisnolinearorqu adratictrendinthe locationparameter withrespecttotime,weconcludethatmodeloneissuperior tomodelstwoandthree. Model ParameterEstimates -2LogL Max.Lik ^ =1764.454,^ =1314.909 1146.77 Lin.Trend ^ a =2147.69, ^ b =-11.14,^ 1145.21 Quad.Trend ^ a =2276.37, ^ b =-23.42,^ c =0.19,^ 1145.07 Table7.2:Log-likelihoodEstimatesforMLLinearandQuadr aticTrendModels 113

PAGE 126

Model4:TheJackknifeModelInthismodel,thejackknifemethodisusedtodecreasethest andarderrorofthemaximumlikelihoodestimatesofthelocationandscaleparamet ers.Themethodderives itsnamefromthewayitisimplementedandwasrstintroduce dbyM.H.Quenouille(1949),andfurtherdevelopedbyJ.Tukey(1958).Th edecreaseinstandard errorisachievedbyconstructingadditionalsamplesetsba sedontheobserveddata values,withoutanyassumptionsastotheestimate'sdistri bution.Recently,Hahnet al.(2002)haveshownthatjackknifebiascorrectedMLestim atesarehigherorder ecient.Ifwelet ^ betheMLestimatorofparameter fromasampleofsizen, theprocedureconsistsofthefollowingsteps: (i)Calculatenestimators ^ i ,whereforeachifrom1ton, ^ i isobtainedusing theexpressiondening ^ eliminatingthei-thobservationsothateach ^ i is calculatedwithasamplesizen-1(forthisreasonthemethod isalsocalledthe leave-one-out method) (ii)Let ^ bethemeanofthesenobservations,i.e, ^ = n X i =1 ^ i n thenthejackknifeestimateof is ^ JK = n ^ ( n 1) ^ (iii)Thejackknifeestimateofthevarianceof ^ isgivenby Var ( ^ JK )= n 1 n n X i =1 ( ^ i ^ ) 2 andthecorresponding(1)100%condenceintervalfor andsucientlylarge 114

PAGE 127

nisapproximatelygivenby ^ JK z = 2 q Var ( ^ JK ) : (7.3.15) Then,jackknifeestimateofthequantilefunction, ^ Q JK isgivenby ^ Q JK =^ JK ^ JK ln ( ln ( F ))(7.3.16) andthejackknifeestimateofthereturnperiodfunctiongiv enby ^ RP JK = 1 1 e e x ^ JK ^ JK : (7.3.17) Table7.3liststhelocationandshapeparameterestimatesu nderthejackknifemodel, alongwiththecorresponding95%condenceintervalsandgo odnessoftp-values. Fromtable7.3wecanseethatthejackknifelocationandscal eparameterestimates areslightlydierentfromtheirMLcounterparts.Fromtabl e7.3wemayconclude thatthejackknifemethodologygiveshigherp-valuesandim provesthegoodnessof tofthemaximumlikelihoodmethod.Asexpected,the95%con denceintervals forthejackknifeparameterestimatesareconsiderablytig hterthanthecorresponding MLcondenceintervals,whichindicatesthatthestandarde rrorsofthejackknife estimatesaremuchlowerthanthoseofMLestimates.Figures 7.5and7.6present thequantileandreturnperiodfunctionsforthejackknifem odel. ^ jk ,95%CI ^ JK ,95%CI K-S 2 A-D 1757.11,(1730.8,1783.5) 1322.76,(1296.8,1348.8) 0.4891 0.5977 0 : 1
PAGE 128

0.2 0 1 0.8 80006000 0.6 40002000 0 0.4 Jackknife Quantile Figure7.5:StreamFlowQuantileFunctionUnderJackknife 60 12000 50 10000 4030 8000 2010 6000 0 4000 2000 0 RP Under Jackknife Figure7.6:StreamFlowReturnPeriodFunctionUndertheJac kknifeModel Model5:ABayesianModelAnalternativewaytotakestatisticaluncertaintiesintoa ccountistoregardthe locationandscaleparametersasbeingrandomquantitiesra therthandeterministic quantities.Onthebasisoftheobservedannualmaximumstre amrowlevels,the priordensityoftheserandomquantitiescanbeupdatedtoth eposteriordensityby usingBayes'theorem.Inordertodescribetheapriori'lack ofknowledge,'weuse thenon-informativeJereyspriorforthelocationandscal eparametersofthetwo parameterGumbeldistributiongivenbyequation(7.3.3),b ecauseforthepurposeof roodprediction,wewouldliketheobservationstospeakfor themselves,especiallyin 116

PAGE 129

comparisontothepriorinformation.Thismeansthatthepri ordistributionshould describeacertain'lackofknowledge,'orinotherwords,sh ouldbeasvagueas possible.Thederivationoftheexpressionforthequantile functionundertheGumbel distributionwasperformedinchapter5underLindley'sapp roximationprocedure. Assumingthatthelocationandscaleparameters and behaveasrandomvariables withajointPDF ( ; ),theBayesquantilefunctionestimate ^ Q B foradesired quantilequnderJereysnon-informativepriorandsquaree rrorlossgivenby: ^ Q B = ^ Q + P 2 u 2 22 + 1 2 ( L 30 2 11 + L 03 u 2 2 22 + L 21 u 2 11 22 + L 12 22 11 ) ; (7.3.18) evaluatedattheMLEofQ, and ( ^ Q ,^ and^ ),where L 2 ; 0 = 1 2 n X i =1 e x i : L 0 ; 2 = n 2 [ n X l =1 2( x i )[1 e ( x i ) ]] 1 3 +( n X i =1 e ( x i ) ( x i ) 2 ) 1 4 : L 3 ; 0 = 1 3 n X i =1 e x i : L 0 ; 3 = 2 n 3 +6[ n X i =1 ( x i )[1 e ( x i ) ]] 1 4 +6[ n X i =1 e ( x i ) ( x i ) 2 ] 1 5 1[ n X i =1 e ( x i ) ( x i ) 3 ] 1 6 : L 21 = n X i =1 e ( x i ) ] 2 3 [ n X i =1 e ( x i )( x i ) ] 1 4 : L 12 = 2 3 ( n n X i =1 e x i )+ 4 4 X ( x i ) e x i 1 5 n X i =1 ( x i ) 2 e x i : 117

PAGE 130

and 11 =( L 20 ) 1 and 22 =( L 02 ) 1 P 1 =0 P 2 = 2 3 andu 2 = ln( ln(1 q )) : Figure7.7presentsthequantilefunctionsunderthejackkn ifeandBayesmodels.The graphsarealmostidentical.Ouranalysisfromchapter5con cludedthat,asexpected, theBayesestimateunderJerey'snon-informativeprioris always"closer"tothetrue stateofnaturethanitsMLcounterpart,sowewillassumetha titisthemodelthat providestheclosestt. 0.2 00.8 0.6 4000 80002000 0.4 0 1 6000 Jackknife Quantile Bayes Quantile Figure7.7:StreamFlowQuantilefunctionUnderJackknifea ndBayesModels Model6:TheNon-parametricKernelDensityModelFollowingtheresultsfromchapter2,weestimatethekernel densitycumulativedensityfunctionusingthevestepprocedure.Thisanalysisis basedontheassumption thatnoparametricmodelcanbefoundtomodelourdata.Inord ertomakeour non-parametricanalysismoreaccurate,weperformedthelo gtransformationofthe data.Thevestepprocedureisgivenasfollows: (i)Chooseapositivenumber h 1 andanintegerk. 118

PAGE 131

(ii)Forh= ih 1 k ,i=1,2,...kcalculatethecorresponding ^ F n anddisplaytheir graphs. (iii)Ifthesekgraphslookalmostthesame,chooseabigger h 1 andgobacktostep 1. (iv)Find i suchthatthegraphsbefore i lookverysimilarandthegraphsafter i lookquitedierentfrombefore. (v)Chooseanyh= ih 1 k ,i
PAGE 132

gure7.8illustratesthatthekerneldensityestimatesoft hereturnperiodfunction arehigherthantheparametriconesforthestreamrowhigher than6,000cfs.Since theAnderson-Darlingtestindicatedthattheparametricmo delswerenotasclosein therighttailasdesirable,wetestedforevidencethatthek erneldensityestimates ofthereturnperiodfunctionprovidemorerealisticvalues intherighttailofthe distribution,bycomparingeachwiththeEmpiricaldistrib utionfunctionofthedata ^ F e .Table7.5liststhedierencebetweentheEmpiricalcumula tivedensityfunction valuesforthetop5valuesandeachofthemodel'scumulative distributionfunctions. Asthedierencesintable7.5indicate,thekerneldensitym odeloverestimatesthe quantilesandistheclosesttotheEmpiricalCDFatthesamet ime,suggestinga heavierrighttailinthedistributionthantheGumbelproba bilitydistributionmodels canhandle. Kernel OptimalBandwidth Kolmogorov-Smirnov 2 Epanechnikov 0.85 p=0.18 p =0 : 2 Table7.4:GoodnessofFitP-valuesfortheKernelDensityMe thodfortheTopKerneland OptimalBandwidth 12000 10000 8000 500 6000 400 4000 300200 2000 100 0 0 Bayes Return Period Jackknife Return PeriodKernel Return Period Figure7.8:ReturnPeriodFunctionUnderJackknife,Bayes, andKernelDensityModels 120

PAGE 133

Value ^ F e ^ F jk ^ F e ^ F B ^ F e ^ F n 12600 -0.00275 -0.0026 0.0012 7750 -0.0051 -0.0055 0.0034 6410 -0.0025 -0.0024 0.0020 5920 -0.0050 -0.0049 0.0044 5890 -0.002 -0.0019 0.0008 5330 -0.0014 -0.0011 0.013 5130 -0.02 -0.02 0.017 4880 -0.03 -0.028 0.09 Table7.5:DierencesBetweentheEmpirical,andJackknife ,Bayes,andKernelDensity CDFEstimatesfortheTopEightTailValues 7.4ModelComparisonandRecommendation Theprimaryobjectiveofanextremevalueanalysisisoftenp rediction,i.e.giventhe historyofannualmaximastreamrow(andotherpossibleexte rnalinformation),the maininterestisinestimatingtheextremalcharacteristic soversomefutureperiod oftime.Thereturnperiodfunctionisdenedastheaveragep eriodofreturnof theextremeeventanditcanindicatehowlongitwilltakefor therecurrenceofa particularstreamrowofinterest.So,amongthesixmodels, whichshouldbeused tomakeapredictionastohowoftenaparticularstreamrowwi llreturn?Figure7.8 suggeststhatkerneldensitymodeloverestimatestheretur nperiodwhencompared toboththejackknifeandBayesianmodelsaroundthemajorro odstagelevels.As table7.5suggests,thisoverestimationprovidesforbette rmodelinginthetailsfor datavalueslargerthanorequalto4880cfs,eventhoughthej ackknifeandBayes modelsperformedbetteroverall.Thisisfurtheremphasize dbytable7.6,which liststhevaluesofthemajorquantilesforthetopvemodels consideredandthe correspondingreturnlevels.Sincethereturnperiodisgiv enbyRP= 1 1 F ,wecan usetable7.6tomakepredictions.Forexample,since 1 1 0 : 995 =200,wemaythinkof thestreamrowlevelsrepresentedby ^ Q (0 : 995)asthelevelsweexpecttooccurevery 121

PAGE 134

200years.Fromtable7.6wecanseethatthekerneldensityqu antilesaresignicantly lowerthanthelowerquantilesandsignicantlyhigherthan theupperquantilesof boththejackknifeandBayesquantileestimates.Thereturn periodandquantile functionsunderthejackknifeandBayesianmodelsshouldbe preferredoverall,but thekerneldensitymodelshouldbeusedtopredictreturnper iodsthatcorrespondto roodlevels. Method ^ Q (0 : 1) ^ Q (0 : 3) ^ Q (0 : 5) ^ Q (0 : 9) ^ Q (0 : 95) ^ Q (0 : 99) ^ Q (0 : 995) ^ Q (0 : 999) Max.Lik 667.78 1520.37 2246.38 4723.48 5669.99 7813.23 8727.96 10846 Jackknife 653.89 1511.57 2241.92 4733.81 5685.97 7842 8762.2 10893.75 Bayes 606.31 1488.76 2240.14 4803.79 5783.37 8001.5 8948.2 11141.13 Kernel 621.7 1332.61 2043.49 5146.24 6172.66 8561.3 10290.13 12914 Table7.6:MajorQuantilesfortheAnnualPeakStreamFlowUn dertheTopFourModels 7.5Conclusion Inthepresentstudy,weconductedbothaparametricandnonparametricestimation ofthequantileandreturnperiodfunctionsfortheannualma ximumstreamrowofthe HillsboroughRiver.Thequantileandreturnperiodfunctio nsareusefulinhydrology, astheycanbeusedtomakepredictionsaboutthefuturerecur renceofpotentially roodcausingstreamrow.Afterexaminingsixdierentmodel s,basedonthegoodnessofttests,andresultsfrompreviouschapters,wefoun dthatjackknifeandBayes methodsprovidedestimatesofbestt.Underthemethodofma ximumlikelihood wefoundnoevidenceoflinearorquadratictrendfortheloca tionparameter.We implementedthejackknifeprocedureandshowedthatitredu cedthestandarderror andprovidedatightercondenceintervalfortheparameter sthanthemaximumlikelihoodmethod.Sincethechoiceoftheunderlyingdistribut ionfunction,inourcase theGumbelprobabilitydistribution,hasaninherentarbit rarinessassociatedwithit, weemployedthemethodofnonparametrickerneldensityesti mationtoderivethe 122

PAGE 135

estimatesofquantileandreturnperiodfunctionsunderthe vestepprocedure.The useofnonparametricdensitieseliminatestheneedforsele ctingaparticulardistributionandthepotentialbiasassociatedwithawrongchoice.N on-parametricallywe usedtheve-stepproceduretoestimatetheoptimalbandwid thandndthemostaccurateestimates.However,weobservedthatkerneldensity estimatesofthequantile andperiodfunctionsprovidedabettertintheextremerigh ttailofthedistribution. Werecommendthatkerneldensityestimationunderoptimalb andwidthbeusedto remedyunderestimationinextremetails,whichmaybeprese ntwhenapplyingthe Gumbelprobabilitydistributionmodel. 123

PAGE 136

WaterYear Date GaugeHeight(ft) PeakAnnualStreamFlow(cfs) 1940 1940-02-18 4.84 724 1941 1941-04-04 12.65 4230 1942 1942-03-03 8.72 1760 1943 1943-08-31 10.23 2350 1944 1944-08-15 6.8 1200 1945 1945-07-26 13.3 5330 1946 1946-08-02 9.4 2010 1947 1947-09-19 13.71 5920 1948 1948-01-25 12.02 3600 1949 1949-08-28 12.9 4620 1950 1950-09-07 13.8 5890 1951 1951-09-19 6.18 1060 1952 1952-03-27 7.6 1420 1953 1953-09-28 12.66 4310 1954 1954-07-27 9.08 1890 1955 1955-09-10 6.93 1240 1956 1956-09-09 6.1 1040 1957 1957-08-07 9.55 2070 1958 1958-02-27 10 2260 1959 1959-03-20 13.1 4880 1960 1960-03-18 15.33 12600 1961 1961-10-11 6.3 1210 1962 1962-08-25 11.2 2940 1963 1963-08-22 6.34 1220 1964 1964-09-11 11.9 3500 1965 1965-08-09 11.85 3460 1966 1966-08-08 8.99 1980 124

PAGE 137

WaterYear Date GaugeHeight(ft) Stream-row(cfs) 1967 1967-08-17 10.98 2380 1968 1968-07-09 10.12 2360 1969 1969-03-18 8.35 1780 1970 1970-10-04 10.97 2800 1971 1971-09-13 4.37 700 1972 1972-08-17 4.79 789 1973 1973-09-10 6.25 1190 1974 1974-06-27 9.95 2200 1975 1975-08-19 8.84 1920 1976 1976-10-05 9.1 2000 1977 1977-09-19 4.86 802 1978 1978-02-19 8.35 1760 1979 1979-09-30 12.25 3850 1980 1980-08-19 4.84 792 1981 1981-09-18 3.44 453 1982 1982-09-26 6.96 1350 1983 1983-03-18 8.47 1800 1984 1984-12-30 6.57 1240 1985 1985-09-07 11.82 3440 1986 1986-03-16 5.74 1020 1987 1987-03-31 12.41 4060 1988 1988-09-09 13.17 5130 1989 1989-11-24 9.96 2330 1990 1990-07-26 2.93 387 1991 1991-07-15 8.08 1550 1992 1992-09-05 4.4 675 125

PAGE 138

WaterYear Date GaugeHeight(ft) Stream-row(cfs) 1993 1993-10-04 6.72 1180 1994 1994-09-17 11.24 2990 1995 1995-09-13 11.31 3040 1996 1996-10-11 8.01 1500 1997 1997-09-28 12.61 4300 1998 1998-12-28 14.22 7750 1999 1999-08-25 3.07 393 2000 2000-09-18 2.43 289 2001 2001-09-15 11.76 3390 2002 2002-09-25 5.28 826 2003 2003-01-02 12.21 3830 2004 2004-09-07 13.93 6410 2005 2005-07-16 6.61 1140 2006 2006-10-25 3.98 557 Table7.7:HillsboroughRiverAnnualPeakStreamFlowNearZ ephyrhills,FL. HillsboroughCounty,Florida.HydrologicUnitCode031002 05.Latitude2808'59",Longitude8213'57"NAD27.Drainagearea220.00squaremiles.C ontributingdrainagearea 220.00squaremiles.Gagedatum33.28feetabovesealevelNG VD29. 126

PAGE 139

8FutureResearch Inthepresentchapterweproposefutureinvestigationsofa reaswhereourresultscan beimplementedandcomparedtotheoneswehaveobtainedsofa r.Inchapter2we providedanewkernelrankingdierentfromSilverman(1986 )basedontheselection oftheoptimalbandwidthforthereliabilityfunction.Anin herentproblemindensity estimationforfailuredataisthespilloverattheorigin,w hichcanbeaddressedusing non-symmetrickernels.Recenteortshavebeenmadetochoo seasymptoticallyoptimalbandwidthsusingBayesianmethodologyandasymmetrick ernels,liketheinverse Gaussiankernel(seeKulasekeraandPadgett(2006)).Wepla ntoextendthesimple vestepproceduretocompareouroptimalbandwidthwiththe newBayesselected bandwidth.Wealsoplantoinvestigatehowdatalteringae ctstherankingofkernelsinkerneldensityestimation.Thesearethenewdirecti onsthatwarrantfurther investigations.Inchapter3wederivedandcomparedtheord inary(MLandMVU), BayesandEmpiricalBayesestimatesfortheGumbelreliabil ityfunctionwiththeir nonparametrickerneldensityestimatesandfoundthemtobe close.Therehasbeen asignicantattentiononthebivariate(andmultivariate) extremevaluedistributions inrecentyears,inparticulartheGumbelmixedmodel,asfor examplewhenmultiple episodicfailureeventshavecorrelatedmaximumpeaksandt otalfailuretimesper testingphase.Ourunivariateresultswillbeextendedtobi variatecases.Chapter 4focusedonderivingandcomparingBayesGumbelreliabilit yestimates,undersix dierentpriors,includingourownkerneldensityprior,an daxedlossfunction, namelythesquareerrorloss.Sincealossfunctionprovides measureofthenancial consequencesarisingfromawrongestimateofanunknownqua ntity,itschoicedoes 127

PAGE 140

notdependonthetypeofestimationbeingused.Theestimati onitselfisoptimal basedonthetypeoflossfunctionbeingchosen.We,asmostot herauthors,chose tousethesquareerrorlossfunctionbecauseitproducedthe posteriormeanasthe Bayesestimatorandwasalgebraicallyeasytocalculate.In practice,however,an overestimationofthereliabilityfunctionisusuallymuch moreseriousthanunderestimation.Also,anunderestimateofthefailureratefuncti onismoreseriousthanits overestimate,soinadditiontostudyingrobustnessoftheG umbelreliabilityfunctionunderdierentsymmetriclossfunctions,weneedtocon siderasymmetricloss functionssuchas LINEX aswell.Inchapter5wederivedtheBayesestimateof targettimesubjecttospeciedreliabilityunderJerey's non-informativepriorand squareerrorlossandcompareditsperformancewithitsordi nary(ML)counterpart. Usingnumericalstudy,wefoundtheBayesianestimatetobem oreeective.Further researchwillcomparetheBayesestimatesunderotherprior s,andextendourresults tootherextremevaluedistributionssuchasFrechet,Paret o,orWeibull.Inchapter 6weintroducedakerneldensityprocedureinordertostudyt heeectivenessofloss functionsinproducingcloseBayesparameterestimatesand showeditwassuperior tootherthreecommonlyusedmethods.Weintendtoapplythis procedureinthe futurewhenstudyingtherobustnessoflossfunctions.Fina lly,inchapter7weused theGumbelprobabilitydistributiontomodelthemaximuman nualstreamrowof theHillsboroughriver.Weconsideredsixdierentmodels. Weusedkerneldensity estimationtoimprovethetintheextremetails,therefore producingcloserextreme quantileandreturnperiodestimates.Ourfutureeortswil lfocusonusingKDE toconsidertheimprovednonparametricestimatesofotherc ommonlyusedextreme valuedistributions. 128

PAGE 141

References [1]Ahsanullah,M.(1990).Estimationoftheparametersoft heGumbeldistribution basedonthemrecordvalues. Comput.Statist.Quart. 6,231-239. [2]Ahsanullah,M.(1991).InferenceandpredictionoftheG umbeldistributionbased onrecordvalues. PakistanJ.ofStatist. 7(3)B,53-62. [3]AliMousaM.A.M,JaheenZ.F.andAhmad,A.A.(2001).Baye sianestimation, prediction,andcharacterizationfortheGumbelmodelbase donrecords. Statistics 36(1),65-74. [4]Basu,A.P.(1964).Estimatesofreliabilityforsomedis tributionsusefulinlife testing. Technometrics .6:215-219. [5]Bean,S.andTsokos,C.P.(1980).Developmentsinnonpar ametricdensityestimation. Intern.Stat.Review .48:267-287. [6]Benson,M.A.(1968).Unformrood-frequencyestimating methodsforfederal agencies. WaterResourcesResearch .4:891-908. [7]Berger,J.O.(1985).StatisticalDecisionTheoryandBa yesianAnalysis.Second Edition.Springer.NY. [8]Bowman,A.&Azzalini,A.(1997).Appliedsmoothingtech niquesfordataanalyisis.Oxford:ClarendonPress. [9]Burton,P.W.andMarkopoulos,K.C.(1985).Seismicrisk ofcircum-Pacic earthquakes:ExtremevaluesusingGumbel'sthirddistribu tionandtherelationship withstrainenergyrealease. PureandAppliedGeophysics .123:849-866. 129

PAGE 142

[10]Castillo,E.&Hadi,A.(2005)ExtremeValueandRelated ModelswithApplicationsinEngeneeringandScience.Hoboken,NJ:WileySeri esinProbabilityand Statistics. [11]Canavos,G.C.&Tsokos,C.P.(1971).Astudyofanordina ryandEmpiricalBayesapproachtoestimationofreliabilityintheGamma lifetestingmodel. Proc.ofIEEESympos.ofReliability [12]Canavos,G.C.&Tsokos,C.P.(1970).OrdinaryandEmpir icalBayesapproachestoreliabilityintheWeibulllifetest ingmodel. Proc.of16thConf.ontheDesignofExperimentsinArmyResea rch .pp379392. [13]Chen,S.X.(1999).Estimationinindependentobserver linetransectsurveysfor clusteredpopulations. Biometrics .55:754-759. [14]Crelin,G.L.(1972).ThephilosophyofmathematicsofB ayes'equation. IEEETransactionsonReliability .VolR-21,No.3,131-135. [15]DiNardoJ.&TobiasJ.(2001).Nonparametricdensityan dregressionestimation. TheJournalofEconomicPerspectives .15:11-28. [16]Donoho,D.L.,Johnston,I.M.,Kerkyacharin,G.,andPi card,D.(1996).Density estimationbywaveletthresholding. AnnalsofStatistics .24. [17]Drake,A.W.(1966).Bayesianstatisticsforthereliab ilityengineer. Proc.Ann.Sympos.onReliability .pp.315-320. [18]Elnaqa,A.&Abuzeid,N(1993).Aprogramoffrequencyan alysisusingGumbel's method. GroundWater ,31:1021-1024. [19]Evans,R.A.(1969).Priorknowledge,engineersversus statisticians. IEEETransactionsonReliability .Vol18,Editorial. [20]Farago,T.andKatz,R.(1990).Extremesanddesignvalu esinclimatology. WorldMetereologicalOrganization WCAP-14,WMO/TD-No.386. 130

PAGE 143

[21]Farquharson,F.A.K.,Meigh,J.&Sutclie,J.V.(1992) .Regionalroodfrequencyanalysisinaridandsemi-aridareas, J.Hydrol. 138,487-501. [22]Fisher,R.A.andTippett,L.H.C.(1928).Limitingform softhe frequencydistributionofthelargestorsmallestmemberof asample. Proc.oftheCambridgePhilosophicalSoceity W24:180-290. [23]Fortin,V.,Bernier,J.,andBobee,B.(1997).Simulati on,Bayes,andbootstrap instatisticalhydrology. WaterResourcesResearch .33(3),439-448. [24]Fortin,V.,Bernier,J.,andBobee,B.(1997).Rational approachtocomparison ofrooddistributionsbysimulation.. J.ofHydrologicEngineering .2(3),95-103. [25]Galambos,J.(1987).Theasymptotictheoryofextremeo rderstatistics(second edition).Krieger,Melbourne,Florida. [26]Gelman,A.,CarlinJ.,SternH.&RubinD.(2004).Bayesi andataanalysis.New York:Chapman&Hall. [27]Ghosh,J.K.&Ramamoorthi,R.V.(2003).Bayesiannonpa rametrics.NewYork: SpringerVerlag. [28]Glasser,G.J.(1962).MVUEforPoissonprobabilities. Technometrics .4:409418. [29]Godbold,J&TsokosC.P.(1970).Theextremevaluedistr ibutionasalife testimgnmodel.VPI&SUTechnicalReport. [30]Goutis,C.(1997).Nonparametricestimationofamixin gdensityviathekernel method. J.ofAmericanStatisticalAssociation .15:11-28. [31]Gu,C.(1993).Smoothingsplinedensityestimation:ad imenslionessautomatic algorithm. J.ofAmericanStatisticalAssociation .88,495-504. [32]Gumbel,E.J.(1958).Statisticsofextremes.Columbia UniversityPress.NY. 131

PAGE 144

[33]Hall,P.&Marron,J.S.(1987).Estimationofintegrate dsquareddensityderivatives. StatisticsandProbabilityLetters .74:109-115. [34]Hosking,J.R.M.,Wallis,J.R.andWood,E.F.(1985).Es timationofthe generailizedextremevaluedistributionbythemethodofpr obabilityweightedmoments. Technometrics 27:251-261. [35]HougaardP.,PlumA.&RibelU.(1989).Kernelfunctions moothingofinsulin absorptionkinetics. Biometrics .45:1041-1053. [36]Jenkinson,A.F.(1969).Estimationofmaximumroods. WorldMeteor.Org.,Tech.Note .No.98,183-257. [37]Jenkinson,A.F.(1955).Thefrequencydistributionof theannualmaximum(or minimum)valueofmeteorologicalelements. Q.J.RoyalMeteorol.Soc. 81,158171. [38]Jones,M.C.,Linton,O.andNielsen,J.P.(1995).Asimp lebiasreductionmethod fordensityestimation. Biometrika .82:327-338. [39]Jones,M.C.,Marron,J.S.,andSheather,S.J.(1995).A briefsurveyofbandwidth selectionfordensityestimation. J.ofAmericanStatisticalAssoc. 90. [40]Kanda,J.(1993).Applicationofanempiricalextremev aluedistributiontoload models. Conf.onExtremeValueTheoryandAppl. May2-7,NationalInstituteof Standards,Gaithersburg,MD. [41]Katz,R.W.,Parlange,M.B.&Naveau,P.(2002).Statist icsofextremesinhydrology. AdvancesinWaterResources .25:1287-1304. [42]Kirby,W.(1969).Ontherandomoccurrencesofmajorroo ds. WaterResourcesResearch. 5:778-784 [43]Kotz,S.andNadarajahS.(2000).Extremevaluedistrib utions:Theoryand applications.London:ImperialCollegePress. 132

PAGE 145

[44]Kottegoda,N.T.,andRosso,R.(1997).Statistics,pro bability,andreliability forcivilandenvironmentalengineers,McGraw-Hill,NewYo rk. [45]Koutsoyiannis,D.(2003).Ontheappropriatenessof theGumbeldistributioninmodellingextremerainfall.Proc.oftheESFLESCExplor.Workshop,Bologna,IT,Oct.200 3. [46]Koutsoyiannis,D.&Baloutsos,G.(2000).Analysisofa longrecordof annualmaximumrainfallinAthens,Greece,anddesignrainf allinferences. NaturalHazards 22(1),31-51. [47]Kulasekera,K.B.andPadgett,W.J.(2006).Bayesbandw idthselectioninkernel densityestimationwithcensoreddata. J.ofNon-parametricStatistics .Vol18,2, 123-149. [48]Liu,K.andTsokosC.P.(2002).Optimalbandwidthselec tionforanon-parametricestimateofthecummulativedistr ibution function. Int.J.ofAppl.Mathematics .Vol10:No1,33-49. [49]Makov,U.(1994).LossrobustnessviaFisherweighteds quarederrorlossfunction. Insurance,Mathematics&Economics [50]Marron,J.S.(1989).Automaticsmoothingparameterse lection:asurvey. EmpiricalEconomic .13:187-208. [51]Miladinovic,B.&Tsokos,C.P.(2008).Bayesianreliab ilityanalysisfortheGumbelfailuremodel. Intern.J.ofNeuralParallel,andScienticComputing Vol16: No1. [52]Miladinovic,B.&Tsokos,C.P.(2008).Ordinary,Bayes ,empiricalBayes, andnon-parametricreliabilityanalysisforthemodiedGu mbelfailuremodel. FifthWorldCongressofNonlinearAnalystsWCNA July2-July9,Orlando,FL. [53]Miroslava,U.(1991).Theextremevaluedistributiono f5-minrainfalldatain Belgrade. TheoreticalandAppliedClimatology .44,223-228. 133

PAGE 146

[54]Maindonald,J.&BraunJ.(2004).Dataanalysisandgrap hicsusingR,.Cambridge:CambridgeUniversityPress. [55]Miller,I.&MillerM.(1999).MathematicalStatistics .PrenticeHall.NJ. [56]Morrison,J.E.&Smith,J.A.(2001).Scalingpropertie sofrood peaks. Extremes .4(1):5-22. [57]Nadaraya,E.A.(1964).Somenewestimatesfordistribu tionfunction. TheoryProbab.Appl 497-500. [58]Naess,A.(1998).Estimationsoflongreturnperioddes ignvaluesforwind speeds. J.Eng.Mech.A.Soc.CivilEng. .124:252:259. [59]North,M.(1980).Timedependentstochasticmodelofro ods. JournaloftheHydrologicalDivision .ASCE,HY5:649-665. [60]Northrop,P.(2004).Likelihoodbasedapproachestoro odfrequency estimation. JournalofHydrology .292:96-113. [61]Osella,A.M.,Sabbione,N.C.,&Cernados,D.C.(1992). Statisticalanalysisofseismicdatafromnorth-westernandwestern Argentina. PureandAppliedGeophysics .139:277-292. [62]Padget,W.J.(1988).Nonparametricestimationofdens ityandhazardratefunctionswhensamplesarecensored.HandbookofStatistics.El savierSciencePublishers.7:313-331. [63]PowellR.A.&SeamanD.E.(1996).Anevaluationoftheac curacyofkernel densityestimatorsforhomerangeanalysis. Ecology .77:2075-2085. [64]Prescott,P.andWalden,A.T.(1980).Maximumlikeliho odestimationofthe parametersofthegenerlizedextreme-valuedistribution. Biometrika .67:723-724. [65]Press,J.S.(2003).SubjectiveandobjectiveBayesian statistics:Principles,modelsandapplications.Hoboken,NJ:WileySeriesinProbabil ityandStatistics. 134

PAGE 147

[66]Pugh,E.L.(1963).Thebestestimateofreliabilityint heExponentialCase. OperationsResearch .11:57-61. [67]Qiao,H.&TsokosC.P.(1992).Bimodalnonparametricde nsityestimation, WorldCongressofNonlinearAnalysts'92. [68]Ramachandran,G.(1982),Propertiesofextremeorders tatisticsandtheirapplicationtoreprotectionandinsuranceproblems. FireSaf.J. .5:59-76. [69]Rao,C.R.(1973).Linearstatisticalinferenceandits applications(secondedition).JohnWileyandSons.NewYork. [70]Rao,N.M.,Rao,P.P.,&Kaila,K.L.(1997).Therstandt hirdasymptotic distributionsofextremesasappliedtotheseismicsourcer egionsofIndiaand adjacentareas. Geophys.J.Int. .128:639-646. [71]Robbins,H.(1980).AnempiricalBayesestimationprob lem. Proc.NationalAcademyofScienceUSA77 .12:6988-6989. [72]Rohatgi,V.K.&Saleh,A.K.(2001).Anintroductiontop robabilityandstatistics(secondedition).JohnWileyandSons.NY. [73]Rychlik,I.(1996).Fatigueandstochasticloads. ScandinavianJ.ofStatistics 23(4):387-404. [74]Sarda,P.(1990).Estimatingsmoothdistributionfunc tions,nonparametricfunctionalestimationandrelatedtopics. KluwerAcademicPublishers .261-270. [75]Savchuk,V.P.&TsokosC.P.(1996).Bayesianstatistic almethodswithapplicationtoreliability.WorldFederationPublishers. [76]Sastry,S.&Pi,J.I.(1991).Estimatingtheminimumofp artitioningandroor planningproblems. IEEETrans.Comput.AidedDesignIntegr.CircuitsSyst. 10:273-282. 135

PAGE 148

[77]Shibata,T.(1993).Applicationofextremevaluestati sticstocorrosion. Conf.onExtremeValueTheoryandAppl. May2-7,NationalInstituteofStandards,Gaithersburg,MD. [78]Silbergleit,V.M.(1996).Ontheoccurrenceofgeomagn eticstormswithsudden commencements. J.Geomagn.Geoelect. .48:1011-1016. [79]Silverman,B.W.(1986)Densityestimationforstatist icsanddataanalysis. Chapman&Hall.London. [80]Simono,J.S.(1996).Smoothingmethodsinstatistics .Springer.NY. [81]SinhaD.Dey,D.,&MullerP.(editors)(1998).Practica lnonparametricand semiparametricbayesianstatistics.Springer. [82]Sinha,S.K.(1986).Reliabilityandlifetesting.Wile yEasternLtd.,Hastead Press. [83]Suzuki,M.&Ozaha,Y.(1994).Seismicriskanalysisbas edonstrain-energy accumulationsinfocalregion. J.Res.Natl.Inst.Stand.Technol. .99:421-434. [84]Tapia,R.A.&Thompson,J.R.(1978).Nonparametricpro babilitydensityestimation.JohnsHopkinsUniversityPress,Baltimore. [85]Tate,R.F.(1959).Unbiasedestimation:functionoflo cationandscaleparameters. Ann.Math.Statist. .30:341-366. [86]Tsokos,C.P.(1972).Bayesianapproachtoreliability :theoryandsimulation. Proc.ofIEEESympos.ofReliability .pp.78-87. [87]Tsokos,C.P.&NadarajahS.(2003).Extremevaluemodel sforsoftwarereliability. Stoch.Anal.&Applic. .21:719-735. [88]Tsokos,C.P.(1999).OrdinaryandBayesianapproachto lifetestingusingthe extremevaluedistribution. FrontiersinReliabilityAnalysis .IAPOQ:379-395. 136

PAGE 149

[89]Turcotte,D.L.(1994)Fractaltheoryandtheestimatio nofextremeroods, J.Res.Natl.Inst.Stand.Technol. 99(4),377-389. [90]Turcotte,D.L.,andB.D.Malamud,B.D.(2003).Applica bilityoffractalroodfrequencystatistics, Hydrofractals'03 ,Aninternationalconferenceonfractalsin hydrosciences,MonteVerita,Ascona,Switzerland,August 2003,ETHZurich,MIT, UniversitPierreetMarieCurie. [91]VanNoortwijk,J.M.(2004).Bayesestimatesofroodqua ntilesusingthegenerilisedgammaditribution.PublishedinHayaka waY.etal. SystemandBayesianReliability:EssaysinHonorofProf.Ri chardE.Barlow pg 351-374.Singapore:WSP.2001. [92]Wald,A.(1943).Testsofstatisticalhypothesisconce rningseveralparameterwhenthenumberofobservationsislarge.Trans.oftheAmericanMathematicalSociety 54:426-483. [93]Wand,M.P.&Jones,M.C.(1995).Kernelsmoothing.Lond on:Chapmanand Hall. [94]Wilks,D.S.(1993).Comparisonofthree-parameterpro babilitydistributionsforrepresentingannualextremeandpartialduration precipitationseries. W WaterResour.Res. ,29(10),3543-3549. [95]Winter,B.B.(1973).Stronguniformconsistencyofint egralsofdensityestimators. Canad.J.Statistics .1:247-253. [96]Yamoto,H.(1973).Uniformconvergenceofanestimator ofdistributionfunctions. Bull.Math.Statist. .15:69-78. [97]Yue,S.(2005).TheGumbelmixedmodelappliedtostormf requencyanalysis. WaterResourcesManagement .14:377-389. 137

PAGE 150

APPENDICES 138

PAGE 151

AppendixI.PartialListofMajorPublicationsintheFieldo fKernel DensityEstimationinthePastFiveYears 1.Heinz,C.(2008).Clusterkernels:resource-awarekerne ldensityestimatorsover streamingdata.IEEETransactionsonKnowledge&DataEngin eering,Vol.20 Issue7,880-893. 2.Gine,E.(2008).Uniformcentrallimittheoremsforkerne ldensityestimators. ProbabilityTheory&RelatedFields,Vol.141Issue3/4,333 -387. 3.Colubi,A.(2008).2008).Favorabilityfunctionsbasedo nkerneldensityestimationforlogisticmodels:Acasestudy.ComputationalSta tistics&Data Analysis,Vol.52Issue9,4533-4543. 4.Pradlwarter,H.J.(2008).Theuseofkerneldensitiesand condenceintervals tocopewithinsucientdatainvalidationexperiments.Com puterMethodsin AppliedMechanics&Engineering,Vol.197Issue29-32,2550 -2560. 5.WhasooB.(2008).AsimplesegmentationmethodforDNAmic roarrayspots bykerneldensityestimation.ORSpectrum,Vol.30Issue2,2 23-234. 6.Menezes,R.(2008)AKernelvariogramestimatorforclust ereddata.ScandinavianJournalofStatistics,Vol.35Issue1,18-37. 7.Sun,X.(2008).Causalreasoningbyevaluatingthecomple xityofconditional densitieswithkernelmethods.Neurocomputing,Vol.71Iss ue7-9,1248-1256 8.Chen,S.(2008)Anorthogonalforwardregressiontechniq ueforsparsekernel densityestimation.Neurocomputing,Vol.71Issue4-6,p93 1-943. 9.Sigalotti,L.(2008).AdaptivekernelestimationandSPH tensileinstability. Computers&MathematicswithApplications,Vol.55Issue1, p23-50. 10.Li,Y.(2008).Integrationofpriorknowledgeofmeasure mentnoiseinkernel densityclassication.PatternRecognition,Vol.41Issue 1,320-330. 139

PAGE 152

11.Yuean,A.(2007).SemiparametricRegressionwithKerne lErrorModel.ScandinavianJournalofStatistics,Vol.34Issue4,841-869. 12.Moser,B.W.(2007).EectsofTelemetryLocationErroro nSpace-UseEstimatesUsingaFixed-KernelDensityEstimator.TheJournalo fWildlifeManagement,v.71,no.7,2421-6. 13.Herzfeld,T.(2007).Corruptionclubs:empiricalevide ncefromkerneldensity estimates.AppliedEconomicsv.39no.10/12(June/July200 7)p.1565-72 14.Fieberg,J.(2007).Utilizationdistributionestimati onusingweightedkernel densityestimators.TheJournalofWildlifeManagement,v. 71,no.5,1669-75. 15.Feiberg,John.Kerneldensityestimatorsofhomerange: smoothingandthe autocorrelationredherringEcologyv.88no.4(April2007) p.1059-66. 16.Ghosh,A.K.(2006).Classicationusingkerneldensity estimates:Multiscale analysisandvisualization.Technometrics,v48i1,120-13 3. 17.Szymkowiak,A.(2006).Clusteringviakerneldecomposi tion.IEEETransactionsonNeuralNetworks,1045-9227,v17,i1,p256(9). 18.Paraskevi,P.(2005).Anevaluationoftheperformanceo fkernelestimatorsfor graduatingmortalitydata.JournalofPopulationResearch 1443-2447,v22i2, p185(13). 19.Lei,L.&Wu,L.(2005).Largedeviationsofkerneldensit yestimatorin L 1 ( R d ) foruniformlyergodicMarkovprocesses.StochasticProces s.Appl.,no.2, 275{298. 20.Buskirk,T.D.,Lohr,S.L.(2005).Asymptoticpropertie sofkerneldensityestimationwithcomplexsurveydata.J.Statist.Plann.Infer ence128,no.1, 165{190. 140

PAGE 153

21.Sahin,K.H.,Diwekar,U.M.(2004).Betteroptimization ofnonlinearuncertain systems(BONUS):anewalgorithmforstochasticprogrammin gusingreweightingthroughkerneldensityestimation.Ann.Oper.Res.132, 47{68. 22.Zhang,B.(2004).Anoteontheasymptoticnormalityofke rneldensityestimatorsunderrandomcensorship.FarEastJ.Theor.Stat.13,no .1,21-31. 23.Chen,S.,Hsu,Y.(2004).Kerneldensityestimationsfor maximumoftwo independentrandomvariables.J.Nonparametr.Stat.16,no .6,901{924. 24.Ahmad,I.A.,Ran,I.S.(2004)Databasedbandwidthselec tioninkerneldensity estimationwithparametricstartviakernelcontrasts.J.N onparametr.Stat. 16,no.6,841{877. 25.Gin,E.,Koltchinskii,V.,Sakhanenko,L.(2004).Kerne ldensityestimators: convergenceindistributionforweightedsup-norms.Proba b.TheoryRelated Fields130,no.2,167{198. 26.Mugdadi,A.R.;Ahmad,I.A.(2004)Abandwidthselection forkerneldensity estimationoffunctionsofrandomvariables.Comput.Stati st.DataAnal.47, no.1,49{62. 27.Lei,L.;Wu,L.(2004)Theexponentialconvergenceofker neldensityestimator in L 1 for -mixingprocesses.Ann.I.S.U.P.48,no.1-2,59{68. 28.Gin,E.,Koltchinskii,V.,Zinn,J.(2004).Weighteduni formconsistencyof kerneldensityestimators.Ann.Probab.32,no.3B,2570{26 05. 29.Gin,E.,Mason,D.M.(2004).Thelawoftheiteratedlogar ithmfortheintegratedsquareddeviationofakerneldensityestimator.Ber noulli10,no.4, 721{752. 30.deValpine,P.(2004).MonteCarlostate-spacelikeliho odsbyweightedposterior kerneldensityestimation.J.Amer.Statist.Assoc.99,no. 466,523{536. 141

PAGE 154

31.Cao,R.,Jcome,M.A.(2004).Presmoothedkerneldensity estimatorforcensoreddata.TheInternationalConferenceonRecentTrendsa ndDirectionsin NonparametricStatistics.J.Nonparametr.Stat.16,no.12,289{309. 32.Delaigle,A.,Gijbels,I.(2004).Practicalbandwidths electionindeconvolution kerneldensityestimation.Comput.Statist.DataAnal.45, no.2,249267. 33.Hallin,M.,Lu,Z.,Tran,L.T.(2004).Kerneldensityest imationforspatial processes:theL1theory.J.MultivariateAnal.88,no.1,61 {75. 34.Kern,J.W.,McDonald,T.L.,Amstrup,S.C.,Durner,G.M. ,Erickson,Wallace P.(2003).UsingthebootstrapandfastFouriertransformto estimatecondence intervalsof2Dkerneldensities.Environ.Ecol.Stat.10,n o.4,405{418. 35.Ferrando,P.J.(2003).Akerneldensityanalysisofcont inuoustypical-response scales.Educ.Psychol.Meas.63,no.5,809824. 36.Biau,G.(2003).Spatialkerneldensityestimation.Mat h.MethodsStatist.12, no.4,371{390. 37.Clements,A.,Hurn,S.,Lindsay,K.(2003).Mobius-like mappingsandtheiruse inkerneldensityestimation.J.Amer.Statist.Assoc.98,n o.464,993{1000. 38.Lei,L.,Wu,L.,Xie,B.(2003).Largedeviationsanddevi ationinequalityfor kerneldensityestimatorin L 1 ( R d )-distance.Developmentofmodernstatistics andrelatedtopics,89{97,Ser.Biostat.,1,WorldSci.Publ ishing,RiverEdge, NJ. 39.Gin,E.,Koltchinskii,V.,Sakhanenko,L.(2003).Conve rgenceindistribution ofself-normalizedsup-normsofkerneldensityestimators .Highdimensional probability,III,241{253,Progr.Probab.,55,Birkhuser, Basel.. 40.Masry,E.(2003).Deconvolvingmultivariatekernelden sityestimatesfromcontaminatedassociatedobservations.IEEETrans.Inform.Th eory49,no.11, 29412952. 142

PAGE 155

41.Wang,X.M.,Zhao,L.C.(2003).Alawoflogarithmforkern eldensityestimatorswithdirectionaldata.(Chinese)ActaMath.Sinica46, no.5,865{874. 42.Ahmad,I.A.,Mugdadi,A.R.(2003)Analysisofkernelden sityestimationof functionsofrandomvariables.J.Nonparametr.Stat.15,no .4-5,579{605. 43.Kim,C.,Kim,W.,Park,B.U.(2003).Skewingandgenerali zedjackkning inkerneldensityestimation.Comm.Statist.TheoryMethod s32,no.11, 21532162. 44.Hazelton,M.L.(2003)Variablekerneldensityestimati on.Aust.N.Z.J.Stat. 45,no.3,271{284. 45.Bouezmarni,T.,Rolin,J.(2003).Consistencyofthebet akerneldensityfunction estimator.Canad.J.Statist.31,no.1,89{98. 46.Berzin,C.,Len,J.R.,Ortega,J.(2003).Convergenceof non-linearfunctionals ofsmoothedempiricalprocessesandkerneldensityestimat es.Statistics37,no. 3,217{242. 47.Gao,F.(2003).Moderatedeviationsandlargedeviation sforkerneldensity estimators.J.Theoret.Probab.16,no.2,401{418. 48.Alberts,T.,Karunamuni,R.J.(2003).Asemiparametric methodofboundary correctionforkerneldensityestimation.Statist.Probab .Lett.61,no.3, 287{298. 49.Park,B.U.,Jeong,S.,Jones,M.C.,Kang,K.(2003).Adap tivevariablelocationkerneldensityestimatorswithgoodperformanceatbou ndaries.J.Nonparametr.Stat.15,no.1,61{75. 50.Duong,T.,Hazelton,M.L.(2003).Plug-inbandwidthmat ricesforbivariate kerneldensityestimation.J.Nonparametr.Stat.15,no.1, 17{30. 143

PAGE 156

51.Acua,E.,Rojas,A.,Coaquira,F.(2002).Eectoffeatur eselectiononbagging classiersbasedonkerneldensityestimators.Classicat ion,clustering,anddata analysis,161{168. 52.Lemdani,M.,Ould-Sad,E.(2002).Exactasymptotic L 1 -errorofakerneldensityestimatorundercensoreddata.Statist.Probab.Lett. 60,no.1,59{68. 53.Gangopadhyay,A.K.Cheung,K.N.(2002)Bayesianapproa chtothechoiceof smoothingparameterinkerneldensityestimation.J.Nonpa rametr.Stat.14, no.6,655{664. 54.Wu,W.B.,Mielniczuk,J.(2002).Kerneldensityestimat ionforlinearprocesses. Ann.Statist.30,no.5,1441{1459. 55.Blower,G.,Kelsall,J.E.(2002).Nonlinearkerneldens ityestimationforbinned data:convergenceinentropy.Bernoulli8,no.4,423{449. 56.Liebscher,E.(2002)Kerneldensityandhazardrateesti mationforcensoreddata underalphamixingcondition.Ann.Inst.Statist.Math.54, no.1,19{28. 57.Campos,V.S.M.,Dorea,C.C.Y.(2002).Kerneldensityes timation:the generalcase.Statist.Probab.Lett.55,no.2,173{180. 144

PAGE 157

AbouttheAuthor BrankoMiladinovicreceivedhisB.S.inMathematicsin1998 fromWesternWashingtonUniversityandM.S.inStatisticsfromtheUniversit yofSouthernMainein 2001,andM.A.inMathematicsfromtheUniversityofSouthFl oridain2003.He hasinstructedvariousmathematicsandstatisticscourses attheUniversityofSouth Florida,HillsboroughCommunityCollege,andtheUniversi tyofTampa.HecurrentlyworksasabiostatisticianattheMottCancerResear chCenterinTampa, Florida.


xml version 1.0 encoding UTF-8 standalone no
record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd
leader nam Ka
controlfield tag 001 002007059
003 fts
005 20090616123514.0
006 m||||e|||d||||||||
007 cr mnu|||uuuuu
008 090616s2008 flu s 000 0 eng d
datafield ind1 8 ind2 024
subfield code a E14-SFE0002760
035
(OCoLC)401321769
040
FHM
c FHM
049
FHMM
090
QA36 (Online)
1 100
Miladinovic, Branko.
0 245
Kernel density estimation of reliability with applications to extreme value distribution
h [electronic resource] /
by Branko Miladinovic.
260
[Tampa, Fla] :
b University of South Florida,
2008.
500
Title from PDF of title page.
Document formatted into pages; contains 144 pages.
Includes vita.
502
Dissertation (Ph.D.)--University of South Florida, 2008.
504
Includes bibliographical references.
516
Text (Electronic dissertation) in PDF format.
3 520
ABSTRACT: In the present study, we investigate kernel density estimation (KDE) and its application to the Gumbel probability distribution. We introduce the basic concepts of reliability analysis and estimation in ordinary and Bayesian settings. The robustness of top three kernels used in KDE with respect to three different optimal bandwidths is presented. The parametric, Bayesian, and empirical Bayes estimates of the reliability, failure rate, and cumulative failure rate functions under the Gumbel failure model are derived and compared with the kernel density estimates. We also introduce the concept of target time subject to obtaining a specified reliability. A comparison of the Bayes estimates of the Gumbel reliability function under six different priors, including kernel density prior, is performed. A comparison of the maximum likelihood (ML) and Bayes estimates of the target time under desired reliability using the Jeffrey's non-informative prior and square error loss function is studied. In order to determine which of the two different loss functions provides a better estimate of the location parameter for the Gumbel probability distribution, we study the performance of four criteria, including the non-parametric kernel density criterion. Finally, we apply both KDE and the Gumbel probability distribution in modeling the annual extreme stream flow of the Hillsborough River, FL. We use the jackknife procedure to improve ML parameter estimates. We model quantile and return period functions both parametrically and using KDE, and show that KDE provides a better fit in the tails.
538
Mode of access: World Wide Web.
System requirements: World Wide Web browser and PDF reader.
590
Advisor: Chris P. Tsokos, Ph.D.
653
Gumbel
Bayesian
Optimal bandwidth
Target time
Unbiased estimation
690
Dissertations, Academic
z USF
x Mathematics
Doctoral.
773
t USF Electronic Theses and Dissertations.
4 856
u http://digital.lib.usf.edu/?e14.2760