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On the theory of records and applications

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Title:
On the theory of records and applications
Physical Description:
Book
Language:
English
Creator:
Mbah, Alfred Kubong
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla.
Publication Date:

Subjects

Subjects / Keywords:
Order statistics
Sequential order statistics
Lower record values
Best linear unbiased estimator
Lower generalized order statistics
Best linear invariant estimator
Best linear least square prediction
Generalized exponential probability distribution
Power function probability distribution
Gumbel probability distribution
Inverse weibull probability distribution
Half logistic probability distribution
Dissertations, Academic -- Mathematics and Statistics -- Doctoral -- USF   ( lcsh )
Genre:
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Dissertation (Ph.D.)--University of South Florida, 2007.
Bibliography:
Includes bibliographical references.
System Details:
System requirements: World Wide Web browser and PDF reader.
System Details:
Mode of access: World Wide Web.
Statement of Responsibility:
by Alfred Kubong Mbah.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 98 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 - 001969427
oclc - 271676479
usfldc doi - E14-SFE0002216
usfldc handle - e14.2216
System ID:
SFS0026534:00001


This item is only available as the following downloads:


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Order statistics.
Sequential order statistics.
Lower record values.
Best linear unbiased estimator.
Lower generalized order statistics.
Best linear invariant estimator.
Best linear least square prediction.
Generalized exponential probability distribution.
Power function probability distribution.
Gumbel probability distribution.
Inverse weibull probability distribution.
Half logistic probability distribution.
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PAGE 1

by AlfredKubongMbah Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofMathematicsandStatistics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:ChrisP.Tsokos,Ph.D. MarcusMcWaters,Ph.D. KandethodyRamachandran,Ph.D. GangaramLadde,Ph.D. DateofApproval: September13,2007 Keywords:Orderstatistics;Sequentialorderstatistics;Lowerrecordvalues;Bestlinearunbiasedestimator;Lowergeneralizedorderstatistics;Bestlinearinvariantestimator;Bestlinearleastsquareprediction;Generalizedexponentialprobabilitydistribution;Powerfunctionprobabilitydistribution;Gumbelprobabilitydistribution;Inverseweibullprobabilitydistribution;Halflogisticprobabilitydistribution. cCopyright2007,AlfredKubongMbah

PAGE 3

IwanttothanktheDepartmentofMathematicsforgivingmeassistantshipduringmytimeasagraduatestudent.IhavefurthermoretothankDr.GeorgeP.YanevandDr.K.RamachandranfromwhomIhavelearntlotsofthingsaboutMathematicsandStatistics.IwillalwaysbegratefultoDr.McWaters,chairmanoftheDepartmentofMathematicsandStatisticsforhisencouragementandsupport. IwanttothankmyfriendandclassmateOneilLynchforhisundyingfriendshipandsupport.Myfamilyhasalsobeenofgratesupporttomethroughoutmytimeasagraduatestudent.IwanttothanktheCamerooniancommunityinTampaforalltheirhelp,support,interest. Especially,IwouldliketogivemyspecialthankstomywifeDoriswhosepatientloveenabledmetocompletethiswork.

PAGE 4

ListofFiguresiv Abstractv 1Introduction1 1.1OrganizationOfTheStudy...........................3 2PowerFunctionProbabilityDistribution:RecordsandApplication6 2.1Introduction....................................6 2.2PropertiesofthePowerFunctionProbabilityDistribution..........6 2.3SimulationStudy.................................17 2.4Conclusion.....................................19 3RecordValuesfromtheInverseWeibullProbabilityDistribution23 3.1Introduction....................................23 3.2DistributionalPropertiesofInverseWeibullProbabilityDistribution.....24 3.3EstimationofParameter.............................26 3.3.1Estimatingandbforknownk.....................26 3.4Estimationofkwhenandbareassumedknown...............28 3.5SimulationStudy.................................30 3.6Conclusion.....................................31 4TheoryOfRecordsForTheGumbelProbabilityDistribution36 4.1Introduction....................................36 4.2AnalyticFormulation...............................37i

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4.3.1Goodness-of-Fit..............................40 4.3.2EstimationofRecords..........................42 4.4Conclusion.....................................46 5RecordvaluesfromtheHalfLogisticProbabilityDistribution47 5.1Introduction....................................47 5.2AnalyticalFormulationoftheRecordModel..................47 5.3Application....................................51 5.4Conclusion.....................................54 6LowerGeneralizedOrderStatistics55 6.1Introduction....................................55 6.2DistributionalPropertiesOfLowerGeneralizedOrderStatistics.......56 6.3PropertiesofthePowerFunctionProbabilityDistribution..........60 6.4MomentsOfLowerGeneralizedOrderStatistics................67 6.4.1Example:UniformProbabilityDistribution..............68 6.4.2Example:PowerFunctionProbabilityDistribution..........69 6.4.3Example:GeneralizedExponentialDistribution............72 6.5EstimationOfParameters............................78 6.5.1PowerFunctionProbabilityDistribution................78 6.6Conclusion.....................................89 7FutureResearchStudies93 References95 AbouttheAuthorEndPageii

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5.1QQplotofthedatatoverifygoodnessoft...................53iii

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2.2CoecientsfortheBLUEof..........................21 2.3VariancesCovariancesoftheBLUEsofandintermsof2.........22 3.1ExpectedvaluesofStandardInverseWeibullpdf................26 3.2VarianceCovarianceofStandardInverseWeibullpdf.............32 3.3CoecientsoftheBLUEforintermsofb...................33 3.4CoecientsoftheBLUEforbintemsofb....................34 3.5VarianceCovarianceofandbintermsofb...................35 4.1Olympicrecordofwomen's100meterfreestyleswimmingresults.......43 4.2ParameterEstimateFortheOlympicdatafrom1912to2004.........43 4.3ParameterEstimateFortheOlympicdatafrom1912to1980.........44 4.4Predictionofnextrecord.............................44 4.5Averageconcentration(pphm)ofSO2fromLongBeach,California......45 4.6ParameterestimatesoftheSO2datasets....................46 5.1Estimateofb(theBoeing720AirplaneData).................52 5.2Estimateofb(theElectricalInsulation)....................54 6.1CoecientsfortheBLUEof..........................90 6.2CoecientsfortheBLUEof..........................91 6.3VariancesCovariancesoftheBLUEsofandintermsof2........92iv

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AlfredKubongMbah LetX1;X2;:::;Xnbeasequenceofindependentandidenticallydistributedrandomvari-ableswithcumulativedistributionfunctionF(x).DenoteXL(n)=minfY1;X2;:::;Xng,n=2;3;:::.XL(j)isalowerrecordoffXngifandonlyifXL(j)
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Theeasiestwaytoexplainhowtostatisticallydenethetheoryofrecordsisbyexamples.Example 10;12;6;15;20;18;17;5;22;3 ThelowerRecordvaluesare 10;6;5;3:

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Chandler,[17]wasrsttointroducetheconceptofrecordvalues,recordtimesandinterrecordtimesforanalyzingthebreakingstrengthdataofcertainmaterial.Heprovedtheresultthatforanygivenprobabilitydistributionfunctionofarandomvariable,theexpectedvalueoftheinterrecordtimeisinnite.Feller[19]gavesomeexamplesofrecordvalueswithrespecttogamblingproblems. Weproceedtonallyintroducesomebasicdenitionsthatplayacentralroleinthepresentstudy.Denition1.0.1 Thejointprobabilitydistributionfunction,pdff(x1;x2;:::;xn)ofrlowerrecordvaluesXL(1);XL(2);:::;XL(r)fromacontinuouscumulativeprobabilitydistributionfunction,cpdf2

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for
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InChapter4,weintroducetheconceptof"records"asappliedtoaprobabilitydistribu-tionfunctionthatcharacterizerealworldphenomenon.Recordsareobtainedbyobservingsuccessiveminimumormaximumvalues.Theproblemofparametricinferenceforrecord-breakingdatawasintroducedbySamaniegoandWhitaker,[47].Theystudiedthepropertiesofmaximumlikelihoodestimatesofthemeanofanunderlyingexponentialprobabilitydis-tribution.GulatiandPadgett,[26],extendedtheworkofSamaniegoandWhitaker,[47],totheWeibulprobabilitydistribution.TheGumbelprobabilitydistributionplaysamajorroleinanalyzingandmodelingthebehaviorofrandomphenomenonthatoccurinengineer-ing,business,biology,nance,sports(MbahandTsokos,[38]),economicsamongothers,additionalreferences,seeLuoandZhu[34],Coles[18],Gumbel[25],Hoskingetal.[27],Kotz[31].WeshallestablishtheanalyticformulationofrecordsasappliedtotheGumbelprobabilitydistribution/doubleexponentialprobabilitydistribution.Inaddition,weshallillustratetheusefulnessofthesubjectresultsintworealworldapplications,namely,theOlympicrecordofwomen's100meterfreestyleswimmingresultsfrom1912to2004andtheonehourmeanconcentrationofsulphurdioxidefromthecityofLongBeach,Californian.Ouranalysisiscomparedwiththeinitialmodelingofthesetwoapplicationswithsignicantimprovement. InChapter5,weintroducethehalflogisticsprobabilitydistributionandstudiedthemaximumlikelihoodestimatesofthelocationandscaleparametersusingrecordbreakingdata.TheusefulnessoftheresultisillustratedusingthefailuretimesofairconditioningequipmentinaBoeing720airplane. Kamps[29]introducedandgavedetailedtheoryoftheso-calledgeneralizedorderstatis-tics(gos)asauniedapproachtoorderstatistics,recordvalues,andsequentialorderstatistics.InChapter6,weintroducetheconceptoflowergeneralizedorderstatistics(lgos).Burkschatetal.[13],createdaconnectionbetweengosandlgos.Ahsanullah[3]presentedseveraldistributionalpropertiesoflgos.Theproblemofestimatingtheparametersofthepowerfunctionprobabilitydistributionbasedonlowergeneralizedorderstatistics.Wealsostudiedthecoecientsofthebestlinearunbiasedestimators(BLUE)forthelocationandscaleparametersofthePowerFunctionProbabilityDistribution.Wealsostudysomedistri-4

PAGE 15

Usingequation(1.0.1)andletting=0and=1,therstmomentofXL(r)fromthepowerfunctionprobabilitydistributionisgivenbyE(XL(r))=r +1r=br:6

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+2r:(2.2.3) Usingexpressions(2.2.2)and(2.2.3),wecomputethevarianceofXL(r)tobeVar(XL(r))= +2r +12r= +1r+1 +1r=arbr;(2.2.4) wherear=+1 +1r: +1s+1 +1r(2.2.5)=bsar: Webeginwiththefollowingcharacterizationofthesubjectpdf,MbahandTsokos,[38].Theorem2.2.1 XhasapowerfunctiondistributionwithF(x)=x,007

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forsomerands,1r1,thatis,fr;s(u;v)=1 (r)(sr)[ln(F(u))]r1[ln(F(u))ln(F(uv))]sr1uf(u)f(uv) (r)(sr)(ln(u))r1(ln(v))sr12(uv)1: Toprove(2)implies(1),using(1.0.4)wehavethatthejointpdfofUandV,fr;s(u;v)=1 (r)(sr)[ln(F(u))]r1[ln(F(u))ln(F(uv))]sr1uf(u)f(uv) Using(1.0.2)and(1.0.4),theconditionalpdfofVjU=u,isgivenbyfVjU(vjU=u)=f(xr;xs) (sr)lnF(uv) for0
PAGE 18

forallu,00:(2.2.10) Thefollowingtheorem,MbahandTsokos,[38]givesfurthercharacterizationofthepowerfunctionprobabilitydistribution.Theorem2.2.2 XhasapowerfunctiondistributionwithF(x)=x,002. forsomerands,1r
PAGE 19

Weobservefrom(1.0.3)thatthecdfofXL(sr)is1ln(F(x))(sr): (2)implies(1). IfVandXL(sr)areidenticallydistributed,thenusingequations(1.0.3)and(2.2.11),wehaveFV()FXL(sr)()=Z101ln(F(u) SinceFbelongstoclassC,weobtainfromtheaboveexpression,(2.2.12)F(uv)=F(u)F(v)(2.2.13) forallu,00 Thefollowingtheorem,MbahandTsokos,[38],identiestheanalyticalestimatoroftheparametersforandwhenisknown.Theorem2.2.3

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9(3r11);

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2(+2)D0+(+1)2(+2)+D0; 2(+2)D0+(+1)2(+2)+D0;

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+1s1xr: +1sxr:

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+1r; +1r+1: xi: Dierentiating(2.2.14)withrespecttoandequatingtozerogivestheMLEestimateof,^MLEtobe^MLE=r

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Observethatthemethodofmaximumlikelihoodperformsbetterthanthemethodofmomentsestimate. UsingTable2.1for=1,n=5,andr=1;:::;5weobtaintheBLUEfor:^=6:130:0255:980:0255:200:0755:060:225+5:031:35=4:962; UsingTable2.2givenbelowfor=1,n=5,andr=1;:::;5wehavethattheBLUEforis^=6:132:05+5:980:05+5:200:15+5:060:455:032:7=2:335; Thestandarderror,S.E.areobtainedfromTable2.3tobeS:E:(^)=p

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21-0.50000-1.00000-2.00000-2.50000-3.00000-3.50000-4.00000-5.00000221.500002.000003.000003.500004.000004.500005.000006.0000041-0.01613-0.07692-0.28571-0.41391-0.55102-0.69433-0.84211-1.1467942-0.03226-0.07692-0.14286-0.16556-0.18367-0.19838-0.21053-0.2293643-0.16129-0.23077-0.28571-0.29801-0.30612-0.31174-0.31579-0.3211441.209681.384621.714291.877482.040822.204452.368422.6972551-0.00321-0.02500-0.133331.69481-0.29779-0.39232-0.49231-0.7038352-0.00641-0.02500-0.066670.11792-0.09926-0.11209-0.12308-0.1407753-0.03205-0.07500-0.133330.21226-0.16544-0.17614-0.18462-0.1970754-0.16026-0.22500-0.266670.38208-0.27574-0.2768-0.27692-0.2759551.201921.350001.60000-2.407081.838241.957352.076922.3175761-0.00064-0.00826-0.06452-0.11176-0.16863-0.23304-0.30332-0.456862-0.00128-0.00826-0.03226-0.0447-0.05621-0.06658-0.07583-0.0913663-0.00640-0.02479-0.06452-0.08047-0.09368-0.10463-0.11374-0.1279164-0.03201-0.07438-0.12903-0.14484-0.15614-0.16442-0.17062-0.1790765-0.16005-0.22314-0.25806-0.26071-0.26024-0.25837-0.25592-0.25069661.200381.338841.548391.642481.734911.827031.919432.1058371-0.00013-0.00275-0.03175-0.06058-0.09788-0.14227-0.19248-0.306372-0.00026-0.00275-0.01587-0.02423-0.03263-0.04065-0.04812-0.0612673-0.00128-0.00824-0.03175-0.04362-0.05438-0.06388-0.07218-0.0857674-0.00640-0.02473-0.06349-0.07852-0.09063-0.10038-0.10827-0.1200775-0.03200-0.07418-0.12698-0.14133-0.15105-0.15773-0.16241-0.168176-0.16001-0.22253-0.25397-0.25439-0.25175-0.24787-0.24361-0.23534771.200081.335161.523811.602681.678301.752761.827071.9768391-0.00915-0.125-0.73587-1.13927-1.57279-2.0248-2.48874-3.4385292-0.001140.000000.045990.085440.131070.180790.233320.3438593-0.00180.000000.052560.095210.143550.195540.249990.3635094-0.002990.000000.061320.107900.159500.214170.270820.3877395-0.005390.000000.073590.125170.180770.238640.297900.4187596-0.010780.000000.091980.150200.210890.272730.335140.4606397-0.025150.000000.122650.190260.257760.324680.390990.5220498-0.075460.000000.183970.266360.343680.417450.488740.62645991.131861.1251.103811.118711.145591.180791.221851.31555101-0.00685-0.11111-0.70697-1.10779-1.54012-1.99161-2.45541-3.40533102-0.000760.000000.039280.073850.114080.158060.204620.30270103-0.001140.000000.044190.081240.123590.169350.217410.31783104-0.001790.000000.050500.090520.135360.183180.232930.33599105-0.002990.000000.058910.102590.150400.200630.252350.35839106-0.005380.000000.070700.119010.170450.223550.277580.387061107-0.010760.000000.088370.142810.198860.255490.312280.42577108-0.025110.000000.117830.180890.243050.304160.364320.48254109-0.075340.000000.176740.253250.324070.391060.455410.5790510101.130141.111111.060461.063631.080241.106131.138511.21600 20

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n=0:5=1=2=2:5=3=3:5=4=5 20.040000.111110.250.308640.360000.404960.444440.510201.160000.777780.68750.693830.706670.721370.736110.76327-0.12000-0.22222-0.375-0.43210-0.48000-0.52066-0.55556-0.6122430.006670.027780.083330.110230.135000.157480.177780.212590.860000.444440.31250.304940.306670.312280.319440.33469-0.02000-0.05556-0.125-0.15432-0.18000-0.20248-0.22222-0.255140.001290.008550.035710.051100.066120.080340.093570.117020.811610.367520.205360.189040.184220.184750.187870.19708-0.00387-0.01709-0.05357-0.07154-0.08816-0.10329-0.11696-0.1404250.000260.002780.016670.026000.035740.045390.05470.071820.802310.344440.16250.139840.130200.126980.127140.13199-0.00077-0.00556-0.02500-0.03640-0.04765-0.05836-0.06838-0.0861860.000050.000920.008060.01380.020240.026960.03370.046610.800460.337010.143150.115930.102640.096520.094330.09569-0.00015-0.00184-0.01210-0.01932-0.02698-0.03467-0.04213-0.0559470.000010.000310.003970.007480.011750.016460.021390.031260.800090.334550.133930.103550.087550.079160.075080.07358-0.00003-0.00061-0.00595-0.01047-0.01566-0.02116-0.02673-0.0375180.000000.000100.001970.004100.006910.010210.013810.021390.800020.333740.129430.096920.078950.068830.063250.05937-0.00001-0.00020-0.00295-0.00574-0.00922-0.01313-0.01727-0.0256790.001060.010230.036790.047560.056320.063460.069350.078390.035760.022730.043770.053690.061850.068500.073980.08238-0.00129-0.01136-0.03884-0.04967-0.05841-0.06548-0.07127-0.08013100.000750.008420.032140.041870.049790.056240.061540.069670.029660.018520.037700.046760.054200.060270.065250.07287-0.00090-0.00926-0.03374-0.04354-0.05145-0.05784-0.06308-0.07106 22

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KellerandKamath,[30],introducedtheuseoftheInverseWeibullprobabilitydistributionasasuitablemodeltodescribethedegradationphenomenaofmechanicalcomponentssuchasthedynamiccomponents(pistons,crankshaft,etc.)ofdieselengines.TheInverseWeibullprobabilitydistributiondistributionalsoprovidesagoodttoseveraldatasuchasthetimestobreakdownofaninsulatinguid,subjecttotheactionofaconstanttension,seeNelson,[41].TheInverseWeibullprobabilitydistributionhasinitiatedalargevolumeofresearch.Forexample,Carriere,[16],hasusedthisdistributiontomodelthemortalitycurveofapopulation,Mohamedetal.,[40],haveconsideredthesingleandproductmomentsoforderstatisticsfrominverseWeibullprobabilitydistributionanddoublytruncatedinverseWeibullprobabilitydistributions,CalabriaandPulcini,[14],havediscussedthemaximumlikelihoodandleastsquaresestimationsofitsparameters,andCalabriaandPulcini,[15],haveconsideredBayes2-samplepredictionofthedistribution. InthischapterweshallusethetheoryofrecordstoobtainsomedistributionalpropertiesoftheInverseWeibullprobabilitydistribution.WeshallobtainparameterofthisdistributionandweshallpresentcoecientsoftheBLUEsofthelocationandscaleparametersoftheInverseWeibullProbabilityDistribution.23

PAGE 33

xb xkexp(x bk);x>0;b>0;0;k>0(3.2.1)=0;otherwise: (r)Z10xnf(xr)dx=k k TherstmomentofXL(r)isobtainedfromequation3.2.2tobeE(XL(r))=r1 ObservealsothatE(XL(1))=(11=k) (1)E(XL(2))=(11=k)E(XL(1));k>1: i;k>1 ThemomentsfortheinverseWeibullprobabilitydistributionhavebeencomputedandarepresentedinTable3.1fork=1:5;2;2:5;:::;5andr=1;2;:::;10.24

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Fromequations3.2.3and3.2.4,wehavethevarianceofXL(r)tobeVar(XL(r))=r1 Usingequation1.0.4,wehavethemthandnthjointmomentsofXL(r)andXL(s),s>rtobeE(XmL(r);XnL(s))=Z10Zxs0xmrxnsf(xr;xs)dxrdxs=k2 whereI(xr)=Zxs01 k(sr) sm k:(3.2.7) Substitutingequation(3.2.7)inequation(3.2.6)givesE(XmL(r);XnL(s))=Z101 k k(r)Z101 ksm+n k k(r);k>m+n:(3.2.8) ThejointmomentofXL(r)andXL(s)canbeobtainfromequation(3.2.8),bytakingm=n=1tobeE(XL(r);XL(s))=r1

PAGE 35

Table3.2presentscomputedvaluesforthevariance-covariancematricesoftheinverseWeibullprobabilitydistributionfunctionfork=2:5;3;3:5;4;4:5;5,r=1;:::;10,s=1;2;3;4;5;6;7;8;10,fors>r. 12.678941.772451.489191.354121.275991.225421.190151.1642320.892980.886230.893520.902750.911420.919060.925670.9313830.595320.664670.714810.752290.781220.804180.822820.8382540.463030.553890.61950.66870.706820.737160.761870.7823650.385860.484660.557550.612980.656330.691090.719540.7432460.334410.436190.512950.572110.618830.656540.687560.7135170.297250.399840.478750.540330.589360.629180.66210.6897380.268940.371280.45140.51460.56530.606710.641080.6700290.246530.348080.428830.493150.545110.587750.623270.65327100.228270.328740.409770.474890.527810.571420.607880.63876 Table3.1:ExpectedvaluesofStandardInverseWeibullpdf3.3EstimationofParameter

PAGE 36

i2 (i2 (r12

PAGE 37

(r); (1): (2)=(11=k)E(XL(1));28

PAGE 38

2k=E(XL(2))E(XL(3)); 2k=E(XL(1))E(XL(3))(3.4.11) ThenexttermresultsinE(XL(3)) 3k=E(XL(3))E(XL(4)):(3.4.12) Addingequations3.4.11and3.4.12togetherresultsinE(XL(1)) 2k+E(XL(3)) 3k=E(XL(1))E(XL(4)) Hence,continuingthisproceduretotherthlowerrecordvaluewehaveE(XL(1)) 2k+:::+E(XL(r1)) DroppingtheExpectationandsolvingfork,wehaveamoment'sestimate^kMEofktobe^kME=1 xk+1i(3.4.14) Theloglikelihoodofequation(3.4.14)islogf1;2;:::;r(x1;x2;:::;xr)=xkr+rXi=1log(k)(k+1)log(xi):(3.4.15)29

PAGE 39

Solvingequation(3.4.16)iterativelygivesthemaximumlikelihoodestimate^kMLEofk.3.5SimulationStudy UsingTable3.4fork=3:5,r=6,wehavethattheBLUEforbis^b=10:874:80+10:372:04+10:032:85+8:153:52+7:64:116:557:37=6:86:30

PAGE 41

112.373150.84530.439350.270810.184260.13376210.199670.117050.077540.05540.041680.03255220.11980.078030.055380.041550.032420.02604310.083220.052890.036920.027380.021170.01688320.049930.035260.026370.020530.016460.01351330.039940.029380.022610.017970.014630.01216410.048580.032130.023040.017420.013670.01103420.029150.021420.016460.013070.010630.00883430.023320.017850.014110.011430.009450.00794440.020210.015870.012770.010480.008750.00741510.032930.022350.016320.01250.00990.00805520.019760.01490.011650.009370.00770.00644530.015810.012420.009990.00820.006850.0058540.01370.011040.009040.007520.006340.00541550.012330.010120.008390.007050.005990.00514610.024290.01680.012430.009610.007670.00628620.014570.01120.008880.007210.005970.00502630.011660.009330.007610.006310.00530.00452640.010110.00830.006880.005780.004910.00422650.009090.00760.006390.005420.004640.00401660.008370.00710.006030.005150.004430.00385710.018920.013280.009930.007730.006210.0051720.011350.008850.007090.00580.004830.00408730.009080.007380.006080.005080.004290.00368740.007870.006560.00550.004650.003980.00343750.007080.006010.005110.004360.003760.00326760.006520.005610.004810.004140.003590.00313770.006080.00530.004580.003970.003460.00302810.01530.010870.00820.006430.005190.00428820.009180.007250.005860.004820.004030.00342830.007350.006040.005020.004220.003590.00308840.006370.005370.004540.003870.003320.00288850.005730.004920.004220.003630.003140.00273860.005270.004590.003980.003440.0030.00262870.004920.004340.003790.00330.002890.00254880.004640.004130.003630.003180.002790.002461010.010820.007840.005990.004740.003860.00321020.006490.005220.004280.003560.003000.002561030.005190.004350.003670.003110.002670.002311040.00450.003870.003320.002850.002470.002151050.004050.003550.003080.002680.002330.002041060.003730.003310.00290.002540.002230.001961070.003480.003130.002770.002440.002150.00191080.003280.002980.002650.002350.002080.001841090.003110.002850.002560.002280.002020.001810100.002980.002750.002480.002210.001970.00176 Table3.2:VarianceCovarianceofStandardInverseWeibullpdf32

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3-0.39628-0.67253-0.93153-1.18546-1.43740-1.68844-2.28280-2.30989-2.46020-2.65333-2.86643-3.090473.679073.982414.391724.838795.303835.7789040.083900.366280.982842.234834.9339012.28310-1.59977-2.55895-4.10090-6.74035-11.87186-25.00456-2.66628-3.83842-5.74126-8.98713-15.26382-31.255705.182157.031099.8593214.4926523.2017844.9771650.064330.252270.612551.220952.202213.77892-0.83573-1.31959-2.00272-2.96920-4.35933-6.42456-1.39289-1.97938-2.80381-3.95894-5.60485-8.03070-1.89939-2.54491-3.46353-4.75072-6.57961-9.266195.063686.591618.6575111.4579115.3415920.9425260.050990.190330.441150.832471.402382.20284-0.51822-0.81763-1.21523-1.73445-2.40705-3.27690-0.86370-1.22645-1.70132-2.31259-3.09478-4.09612-1.17777-1.57686-2.10163-2.77511-3.63300-4.72629-1.47221-1.89224-2.45190-3.17156-4.08712-5.251444.980916.322858.0289210.1612412.8195716.147907-0.00409-0.01039-0.01766-0.02518-0.03255-0.03958-0.02167-0.02261-0.02255-0.02201-0.02125-0.02041-0.03612-0.03392-0.03157-0.02934-0.02732-0.02552-0.04925-0.04361-0.03899-0.03521-0.03207-0.02944-0.06156-0.05234-0.04549-0.04024-0.03608-0.032720.037100.021740.011470.00436-0.00072-0.004451.135601.141131.144791.147611.149991.152128-0.00262-0.00705-0.01251-0.01839-0.02431-0.03007-0.01889-0.02008-0.02025-0.01992-0.01935-0.01867-0.03149-0.03011-0.02835-0.02657-0.02488-0.02334-0.04294-0.03872-0.03502-0.03188-0.02921-0.02693-0.05367-0.04646-0.04086-0.03643-0.03286-0.02992-0.50563-0.50568-0.50522-0.50468-0.50419-0.503811.060351.083151.098231.108981.117141.123620.594880.564950.544000.528890.517670.5091210-0.00155-0.00440-0.00223-0.01223-0.01648-0.02069-0.01246-0.013720.00410-0.01421-0.01399-0.01364-0.02076-0.020590.00574-0.01895-0.01798-0.01704-0.02831-0.026470.00709-0.02274-0.02111-0.01967-0.03539-0.031760.00827-0.02598-0.02375-0.02185-0.43398-0.447260.83452-0.46255-0.46735-0.471110.957390.99739-2.055341.045851.061641.07420-0.67657-0.679611.26150-0.68186-0.68231-0.68260-0.06098-0.049710.01211-0.03618-0.03183-0.028411.312611.27613-0.075751.228851.213161.20080 Table3.3:CoecientsoftheBLUEforintermsofb.33

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30.574930.916621.214241.494521.765812.031733.104532.979933.066173.224633.416203.62566-3.67946-3.89654-4.28042-4.71915-5.18201-5.657394-0.16832-0.63722-1.56650-3.34699-7.04898-16.909412.648413.997376.126339.7112116.5958034.070334.414015.996058.5768612.9482721.3374542.58792-6.89410-9.35621-13.13669-19.31249-30.88427-59.748845-0.13966-0.47463-1.04811-1.94912-3.33327-5.483031.529542.229873.188934.511826.377099.105902.549233.344814.464506.015768.1991111.382383.476234.300475.514977.218919.6250513.13351-7.41534-9.40052-12.12030-15.79736-20.86797-28.138766-0.11857-0.38113-0.79766-1.39593-2.21862-3.327081.027501.472112.038262.753523.656344.800151.712492.208172.853573.671364.701016.000182.335222.839083.524994.405645.518576.923292.919023.406894.112495.035016.208397.69254-7.87566-9.54513-11.73165-14.46961-17.86569-22.089087-0.00442-0.00922-0.01332-0.01665-0.01934-0.02154-0.00138-0.00090-0.00061-0.00042-0.00030-0.00022-0.00231-0.00136-0.00085-0.00056-0.00038-0.00027-0.00314-0.00174-0.00105-0.00067-0.00045-0.00031-0.00393-0.00209-0.00123-0.00077-0.00051-0.000352.368222.129611.974071.864661.783541.72102-2.35303-2.11430-1.95701-1.84559-1.76256-1.698338-0.00076-0.00149-0.00198-0.00228-0.00244-0.002520.005540.004970.004440.003980.003590.003270.009230.007460.006220.005310.004620.004090.012580.009590.007680.006370.005430.004710.015730.011510.008960.007280.006100.005241.016660.908160.837290.787480.750620.72227-2.54041-2.24859-2.05947-1.92732-1.82996-1.755341.481451.308381.196861.119181.062041.0182810-0.00086-0.00168-0.00223-0.00256-0.00274-0.002830.004910.004520.004100.003720.003390.003100.008180.006780.005740.004960.004360.003880.011160.008710.007090.005950.005110.004480.013950.010450.008270.006800.005750.004971.009680.903960.834520.785540.749200.72119-2.53035-2.24240-2.05534-1.92442-1.82782-1.753711.597181.392191.261501.171241.105321.055140.024040.016360.012110.009470.007710.00647-0.13788-0.09889-0.07575-0.06071-0.05028-0.04269 Table3.4:CoecientsoftheBLUEforbintemsofb.34

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30.59885460.51567010.48442710.47029110.463470.460231.1815770.90670080.78698270.72056050.678560.64971-0.814424-0.6652533-0.6037476-0.5715508-0.55236-0.5399340.419670.571270.807491.19471.919543.723661.205151.432631.830862.502223.768886.92783-0.69477-0.89239-1.20631-1.72127-2.68335-5.0737350.219240.294590.394350.526280.704850.956740.775320.86991.021131.231631.521461.93268-0.40125-0.49781-0.62792-0.7997-1.0311-1.3560460.135950.182530.239290.307420.389190.487990.567070.614520.690060.787840.908381.05556-0.2695-0.32864-0.40135-0.488-0.59119-0.7148370.005680.005050.004440.00390.003440.003040.00780.005240.003760.002840.002220.001780.000362.00E-040.000127.00E-055.00E-053.00E-0580.004960.004480.003990.003530.003130.002780.003280.00220.001580.001180.000920.00074-0.00145-0.00111-0.00087-0.00071-0.00058-0.0004990.003970.003660.00330.002950.002630.002350.003270.002190.001570.001180.000920.00074-0.00136-0.00105-0.00084-0.00068-0.00056-0.00047100.003270.003060.002790.002520.002260.002030.003260.002190.001570.001180.000920.00074-0.00129-0.00101-0.00081-0.00066-0.00055-0.00046 Table3.5:VarianceCovarianceofandbintermsofb.35

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;
PAGE 46

GivenacompleterandomsampleX1;X2;:::;Yn.2. Therstrecord,XL(1),isX1,therstobservation,thatis,XL(1)=X1.Thesecondrecordvalue,XL(2),isobtainedbyobservingtheindependentandidenticallydistributedrandomvariablesXi'ssequentiallyfromX2;:::;Xn.ThenextobservationthatislessthanXL(1)isthesecondrecordobservation,XL(2),andthenumberoftrialstogetXL(2)isK1,forexample,letthenextobservationthatislessthanXL(1)beX5,thenXL(2)=X5andK1=4.Now,X5isastandardforgettingsubsequentrecords.3. TheobservedatawillconsistofXL(1)=x1;K1=k1;XL(2)=x2;K2=k2;:::;XL(r)=xr;Kr=kr,wherefXL(i);1irgistherecordvaluesequenceandfKi;i>0gandKr=1istheinterrecordtimesequence.Notethatbyusingthismethod,thenumberofrecordsobtained(r)willbelessthann,thesizeofthecompleterandomdatasample. Notethattherecordvalueswithouttheinterrecordtimesformwhatisknownasthelowerrecordvalues. Fortherecord-breakingsamplesx1;k1;x2;k2;:::;xr;krdenedabove,wecanwritethelikelihoodfunctionasL(x)=rYi=1f(xi)(1F(xi))ki1;(4.2.2) wheref(x)andF(x)arethepdfandcdfoftherandomvariablesfromwhichtherecordobservationsareobtained.37

PAGE 47

wherezi=(xi)=. Thenegativeloglikelihoodofexpression(4.2.3)isgivenbyf()=logL(z)=rXi=1log()zi+kiezi;(4.2.4) where=(;). Takingthepartialderivativeof(4.2.4)withrespecttoandwehave@f() and@f() Setequations(4.2.5)and(4.2.6)equaltozeroweobtainthemaximumlikelihoodesti-mates^forand^forfortherecordsamples. Thesecondpartialderivativesofequations(4.2.5)and(4.2.6)withrespectto,andaregivenby@2f() 2;(4.2.7)@2f() and@2f()

PAGE 48

^20@rPri=1ki^zie^ziPri=1ki^zie^zir+Pri=1ki^zi2e^zi1A:(4.2.10) Approximatecondenceintervalsandhypothesistestsfor,andcanbefoundbytreating(^;^)asabivariatenormalforlargesampleswithmeanvector(;)andcovariancematrixtheinverseofI(^;^),thatisI(^;^)1. Next,weproceedtoobtaintheestimatesoftheparametersthatareinherentinequations(4.2.5)and(4.2.6)asfollows,forthecompletesampleX1;:::;Xn,fromtheGumbelprob-abilitydensityfunctiongivenbyequation(4.1.1),wecanwritethenegativelog-likelihoodasg()=nXi=1log()i+exp(i);(4.2.11) wherei=(xi)=. Takingthepartialderivativesofequation(4.2.11)withrespecttoand,wehave@g() and@g() Letexpressions(4.2.12)and(4.2.13)equaltozeroandtakingthesecondpartialderivativeswithrespectto,andwehave@2g() 2;(4.2.14)@2g()

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Usingequations(4.2.14),(4.2.15),and(4.2.16)weobtaintheobservedinformationmatrixat(^;^),thatis,I(;),fortheGumbelpdfmodeltobeI(^;^)=1 ^20@nPni=1^ie^iPni=1^ie^in+Pni=1^i2e^i1A:(4.2.17) Thus,wecanusetheestimatesinequations(4.2.12)and(4.2.13)topredictfutureob-servations.Wecanaccomplishthisbyusingthereturnlevels,thatis,F(xs)=1=s;s>r s1:(4.2.18)4.3ApplicationsOfRecords

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WeshallusethetwosidedKolmogorov-SmirnovteststatisticgivenbyDn=supxjFn(x)F(x)j;(4.3.19) whereFn(x)istheempiricalprobabilitydistributionfunction,F(x)istheGumbelproba-bilitydistributionfunctionthatwearetestingtoseeifthegivendatatsitwell.Notethatthestatistic(Dn)isdistributionfree. FortheOlympicdata,usingequations(4.2.12)and(4.2.13),wehavetheestimates^ofand^offortheGumbelpdfequalto78.61and5.66,respectively. UsingtheseestimatesfortheOlympicdataandthoseofRoberts,[45],forthehourlyconcentrationofSO2,weareabletodetermineifthedataactuallyfollowstheGumbelpdf.TheresultsgivenbelowverifythatindeedtheGumbelpdftsthegivendata. Olympic 0.1808 Good-ness-ofttest WehavealsousedtheAkaikeinformationcriterion(AIC)andquantileplottofurtherverifythetofourmodelwiththatofAhsanullah,[5]. TheAICisdenedby2logL+2K;whereListhelikelihoodofthemodelofinterestandKisthenumberofparametersinthemodel.ThesmallerthevalueoftheAIC,thebetterthemodel. AquantileplotisttedandplottedagainsttheexpectedquantiledenedbyF(xi)=(i0:375)=(n+0:25)(Royston[46]),whereFisgivenbyequation(4.1.1).Forexample,tocheckthequantileplotofourmodel,wewouldplottheexpectedquantileversusthettedquantilesusingthemaximumlikelihoodestimatesgivenbyequations(4.2.5)and(4.2.6),andthecdfdenedinequation(4.1.1).41

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andki=1;1;1;1;1;3;1;1;2;1;1;3;1;1;142

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191282.20196860.00192073.60197258.59192472.40197655.65192871.00198054.79193266.80198455.92193665.90198854.93194866.30199254.64195266.80199654.50195662.00200053.83196061.20200453.84196459.50 Table4.1:Olympicrecordofwomen's100meterfreestyleswimmingresults. Table4.2givestheparameterestimatesofourmodel,thatis,usingrecordvaluesversuscompletedataanalysis.Ascanbeobserved,theestimatesareverygood. CompleteData ^ 78.61 ^ 5.66 Table4.2:ParameterEstimateFortheOlympicdatafrom1912to2004 Next,wecompareouranalysiswiththatofAhsanullah,[5].FromthedatapresentedinTable4.1,Ahsanullah,[5]usedthelowerrecords,from1912to1980,thatis,82:2;73:6;72:4;71;66:8;65:9;62;61;59:5;58:59;55:65;54:79 toobtain^=78:74,^=3:97asthebestlinearunbiasedestimates(blue). UsingthemethodpresentdescribedinSection2,weobtainfromTable4.1therecordvaluesandrecordtimesrespectivelyfortheperiod1912to1980asxi=82:2;73:6;72:4;71;66:8;65:9;62;61;59:5;58:59;55:65;54:79 andki=1;1;1;1;1;3;1;1;2;1;1;1:

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Table4.3presentsasummaryoftherecordestimatesobtainedusingthemethodsdescribeaboveversusthoseofAhsanullah,[5]. Ahsanullah(1994) ^ 78.74^ 3.97AIC 94.5 103.61 Table4.3:ParameterEstimateFortheOlympicdatafrom1912to1980 sOurmodelAhsanullah'smodelActualOurmodelAhsanullah'smodel 1354.7154.3954.64+.07-.251454.2353.9154.50-.27-.591553.7453.4653.83-.09-.371653.2853.011752.8652.56 Furthermore,wesummarizewhatactuallyhappenedinthesubjectmatterinthe1992,2000,and2004Olympicgames:

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Max 195647314412133142133264032 471957221920322023181613144125 411958151320122413372032272768 681959203220153681517152020 3219602218232081314913162720 27196125202016101081012161443 4319622013151810110101111147 20196312182721274415101818 27196416103319916254141821 2519651618914810181814121212 181966273325101730131822152523 331967304032108782610401817 401968513018221019222526295040 51196937135514910131733131544 55197023191011151225402520128 401971223620281015205538412625 551972303218273713231921312513 3719731088121116251611281023 281974899138149925111915 34 Table4.5:Averageconcentration(pphm)ofSO2fromLongBeach,California. FromthelastcolumnofTable4.5,thefollowingrecordvaluesandinterrecordtimesareobservedxi=47;41;32;27;20;18;

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^20.5544.3031.50^8.7315.412.34AIC53.6353.99 Table4.6:ParameterestimatesoftheSO2datasets4.4Conclusion

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BalakrishnanandChan,[9],havestudiedthebestlinearunbiasedestimatorofthescaledparameterhalflogisticprobabilitydistributionusingdoubletypeIIcensoredsamples.InthisChapter,weshallstudythetheoryofrecordsforthehalflogisticprobabilitydistribution.Inadditiontodevelopingtherecordestimatesoftheparameters,weshallillustratetheirusefulnessbyapplyingtheresultstothefailuretimedataofBoeing720airplane.ThisdatahasbeeninitiallyanalyzedbyBalakrishnanandChan,[9].5.2AnalyticalFormulationoftheRecordModel b b2;(5.2.1)47

PAGE 57

Thecumulativeprobabilitydistributionfunctionofequation5.2.1isgivenbyF(x)=1ex b b:(5.2.2) Fortherecord-breakingsamples,x1;k1;:::;xr;kr,usingequations(4.2.2),(5.2.1)and(5.2.2),thelikelihoodfunctionforthehalflogisticprobabilitydistributionfunctionisgivenbyL(!)=rYi=11 1+e!i2e!i where!i=(xi)=b. Forconvenience,thenegativeloglikelihoodfunctionofexpression(5.2.3)isgivenbylogL(!)=rXi=1log(b)+ki!i+(1+ki)log(1+e!i):(5.2.4) Takingthepartialderivativesofequation(5.2.4)withrespecttoandbwehave@(logL(!)) and@(logL(!)) Thesecondpartialderivativeofequations(5.2.5)and(5.2.6)withrespectto,bandbaregivenby@2(logL(!))

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and@2(logL(!)) Equating(5.2.5)and(5.2.6)tozero,weobtainthemaximumlikelihoodestimates^and^bforandb,respectively,andequations(5.2.8)and(5.2.9)become@2(logL(!)) and@2(logL(!)) Usingequations,(5.2.7),(5.2.10),and(5.2.11)weobtaintheobservedinformationmatrixI(;b),at(^;^b),forthehalflogisticpdfmodeltobeI(^;^b)=1 ^b20@Pri=1(1+ki)e!i NotethattheinverseofI(^;^b)fromequation5.2.12givesthevariancecovariancematrixof^;^b. Next,weproceedtoobtaintheestimatesoftheparametersthatareinherentin(5.2.1)asfollows:ForthecompletesampleX1=y1;:::;Xn=yn,fromthehalflogisticprobabilitydensityfunctiongivenby(5.2.1),wecanwritethenegativelog-likelihoodfunctionaslogL()=nXi=1log(b)++2log(1+exp();(5.2.13) wherei=(yi=b.49

PAGE 59

and@(logL()) Letexpressions(5.2.14)and(5.2.15)equaltozeroandtakingthesecondpartialderivativewithrespectto,bandb,wehave@2(logL()) and@2(logL) Usingequations(5.2.16),(5.2.17),and(5.2.18)weobtaintheobservedinformationmatrixat(^;^b),thatis,I(;b),forthehalflogisticpdfmodeltobeI(^;^b)=1 ^b20@Pni=12e^i

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Thefollowingrecordvaluesandrecordtimescanbeobtainedfromtheabovedata,thatis,xi=74;57;47;29;12 withki=1;1;1;2;1: Table5.1presentsacomparisonoftheparameterestimateofourmodelwiththatofBalakrishnanandChan,[9].Ascanbeobservedfromthetablebelow,eventhoughwehaveareducedsample,ourmodelperformsequallywellwiththemaximumlikelihoodestimate(MLE).51

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MLEusingRecords88.23(20.22)BLUEusingCompletedata90.92(19.66)MLE93.58 Table5.1:Estimateofb(theBoeing720AirplaneData) Thebestlinearunbiasedestimateforbhasbeenobtainedusingthecompletedata,BalakrishnanandChan,[9].BalakrishnanandChan,[9]obtainedtheblue(standarderror)forbtobe50.50(12.68).Usingrecords,wewillobtaintheblueforbandcompareourresultwiththatofBalakrishnanandChan,[9]. Usingequation(5.2.15),weobtainthemaximumlikelihoodestimateforthescaleparam-eterofthehalflogisticpdftobe^b=47:42. TheappropriatenessoftheassumptionofthehalflogisticdistributionfortheabovedataischeckedusingtheQ-QplotgivenbyFigure5.1.InFigure5.1,weplottedthequantilefromthehalflogisticprobabilitydistributionversustheempiricalquantiles.AscanbeobservedfromFigure5.1,thehalflogisticdistributiontsthedataextremelywell.ThevalueofthecorrelationcoecientintheQ-Qplotis0.98.52

PAGE 62

andki=1;3;2;1;3;1 Usingequation(5.2.6)wecalculatedthemaximumlikelihoodestimatesandtheirstandarderrorsofbtobe^bmle=52:22withastandarderrorof14.22.TheresultsaresummarizedinTable5.2below. Table5.2presentsestimatesofthedatausingourmodel,withthoseofBalakrishnanandChan,[9].Ascanbeobserved,wehavesuccessfullypresentedanothermethodtoestimatetheparameterofthescalehalflogisticprobabilitydistribution.Thesubjectmethod,records53

PAGE 63

MLEusingRecords52.23(14.22)BLUEusingCompletedata50.50(12.68)MLE47.42 Table5.2:Estimateofb(theElectricalInsulation)andinterrecords,givegoodresultwitheasiercomputationalprocessandsmallersamples.5.4Conclusion

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Inthischapter,wewillconsiderlowergeneralizedorderstatistics(lgos).Lgosisageneralizationoforderstatistics,thatis,sortingthevaluesofrandomvariablesinde-creasingorder;lowerrecordvalues.Foraconnectionbetweengeneralizedorderstatisticsandlowergeneralizedorderstatistics,seeBurkschatetal.,[13]. SupposefX(1;n;m;k);:::;X(n;n;m;k)g,wherek1,arerealnumbers,denotenlgosfromanabsolutecontinuouscumulativedistributionfunctionwithcdfF(x)andcorre-spondingprobabilitydensityfunction(pdf)f(x)=dF(x)=dx,thejointpdff1;:::;n(x1;:::Xn)ofnlgosX(1;n;m;k);X(2;n;m;k);:::;X(n;n;m;k)isf12:::n(x1;x2;:::;xn)=kn1Yj=1jn1Yi=1(F(xi))m(F(xn))k1f(x1)f(xn)(6.1.1) forF1(1)x1x2:::xnF1(0);m>1,r=k+(nr)(m+1);r=1;2;:::;n1;k1; Observethatequation(6.1.1)hasasspecialcaseorderstatisticsifweletk=1andm=0,whilelowerrecordvaluesarespecialcasesofequation(6.1.1)ifweletk=1and55

PAGE 65

Integratingoutx1;:::;xr1;xr+1;:::;xnfrom(6.1.1),weobtainthepdffr;n;m;kofX(r;n;m;k)asfr;n;m;k(x)=8<:cr1 wherecr=rYi=1i; forF1(0)
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(r)Zx0ur1eudu; (p+q): m+1+n1)(k m+1+n2):::(k m+1+nr)(r andZ10wa1(1wb)r1dw=1 Usingequation(6.1.2),andform>1,Fr;n;m;k(x)=cr1 m+11(1t)r1dt=cr1B(r;r (r)(m+1)rI1(x)r (r)(m+1)r(r+r

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(r)Zkln(F(x))xtr1etdt=12(r): Proof.Forconvenienceweshallgivetheproofforabsolutecontinuousdistributions.Itcaneasilybeshownfromequation(6.1.1)thatthejointpdfofslgosfX(1;n;m;k);X(2;n;m;k);:::;X(r;n;m;k);X(s;n;m;k)gforr
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ObservethatthejointpdfofrlgosfX(1;n;m;k);X(2;n;m;k);:::;X(r;n;m;k)gisgivenbyf1;2;:::;r;m;n;k(x1;x2;:::;xr)=cr1"r1Yj=1(F(xj))mf(xj)#(F(xr))r1f(xr):(6.2.8) Usingequations(6.2.7)and(6.2.8),weobtaintheconditionalpdfofX(s;n;m;k)givenX(1;n;m;k)=x1;X(2;n;m;k)=x2;:::;X(r;n;m;k)=xrasf1;2;:::;r;m;n;k(xsjx1;x2;:::;xr)=f1;2;:::;r;s;m;n;k(x1;x2;:::;xr;xs) ForF1(0)
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Xisuniformlydistributedrvin(0;1)2. XhasapowerfunctiondistributionwithF(x)=x,002. whereWr+1isindependentofX(r;n;m;k)andthepdfofWr+1isfr+1(w)=r+1wr+11;00: XhasapowerfunctiondistributionwithF(x)=x,002. forsomerands,1r
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Thus,ifXhascdfF(x)=x,thenUandVareindependent. Toprove(2)implies(1),usingequation(6.1.3)wehavethejointpdfofUandV,form>1,thatis,fr;s;n;m;k(u;v)=cs1 Usingequations(6.1.2)and(6.3.11),theconditionalpdfofVjU=u,form>1,isgivenbyfVjU(vjU=u)=cs1

PAGE 71

SinceUandVareindependent,wecanwriteIF(uv0) forallu,00:(6.3.15) Form=1,thejointpdfofX(r;n;m;k)andX(s;n;m;k),r
PAGE 72

Integratingequation(6.3.16)withrespecttovfrom0tov0,weobtainFVjU(vjU=u)=ks (sr)ZklnF(uv0) Lettingu!1,wehavethatklnF(uv0) forallu,0
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XhasapowerfunctiondistributionwithF(x)=x,002. forsomerands,1r1,wehavefU;V(u;v)=fVjU=u(ju)f(u) using(6.3.13)wecanwritetheprobabilitydistributionfunctionasFV()=Z10Z0fV=U=u(t=u)f(u)dtdu(6.3.19)=Z10nI(F(u)

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Now,weshallprovethat(2)implies(1).IfVandX(sr;ns;m;k)areidenticallydistributed,thenusingequation(6.3.13)andLemma6.2.2,wecanwriteZ10nI(F(uv) Uponsimplication,weobtainfromequation(6.3.20)Z10nhI(F(uv) SinceFbelongstoclassC,theclassofallcontinuousfunctions,weobtainfromequation(6.3.21)F(uv)=F(u)F(v);(6.3.22)65

PAGE 75

Thenon-zerosolutionofequation(6.3.22)withtheconditionsthatF(x)isaprobabilitydistributionfunctionwithF(0)=0;andF(1)=1; Form=1,(1)implies(2),wehavethatFV()=Z10FV=U=u(=u)f(u)du(6.3.23)=Z101kln(F(u) Now,toshowthat(2)implies(1),IfVandUareidenticallydistributed,thenusingequation(6.3.17)andLemma6.2.2,weobtainFV()=Z101kln(F(u)

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SinceFbelongstoclasstheC,weobtainfromequation(6.3.25)F(uv)=F(u)F(v);(6.3.26) forallu,00: Fork=1andm=1,weobtainedacharacterizationofthepowerfunctiondistributionbasedontheidenticaldistributionoflowerrecordvaluesXL(s)jXL(r)andXL(sr).6.4MomentsOfLowerGeneralizedOrderStatistics Settingj=1andj=2,inequation(6.4.27),weobtaintherstmoment(r;n;m;k)andsecondmoments(2r;n;m;k),respectivelyofX(r;n;m;k).Thevariance2r;n;m;kofX(r;n;m;k)is67

PAGE 77

Thejoint(i;j)thmomentofX(r;n;m;k)andX(s;n;m;k)isgivenbyijr;n;m;k=E(X(r;n;m;k))i(X(s;n;m;k))j=Z1Zxuivjfr;s;n;m;k(u;v)dudv;(6.4.29) withfr;s;n;m;kgivenbyequation(6.1.3).Thecovariance2r;s;n;m;kofX(r;n;m;k)andX(s;n;m;k)isexpressedas2r;s;n;m;k=2r;s;n;m;kr;n;m;ks;n;m;k(6.4.30) Weproceedtostudysomeexamplesbasedonthedevelopedtheory.6.4.1Example:UniformProbabilityDistribution

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(m+1)(n+2): (m+1)r(r)(r+1

PAGE 79

(m+1)r(r)(r+2 (1+2):::(r+2)crr (nr+1)(n+1+1

PAGE 80

(nr+1)(n+1+2 +1r

PAGE 81

wherex>;<<1;>0;>0. Withoutlossofgenerality,wewilltake=0,and=1.Substitutingequation(6.4.38)intoequation(6.1.2),weobtainforr=1,f1;n;m;k(x)=1(1ex)1ex:(6.4.39)Usingequations(6.4.27),and(6.4.38)andthesubstitutionw=1ex,therstmomentofrlgosfromtheGEDisgivenbyE(X(r;n;m;k))=cr1

PAGE 82

UsingtheMaclaurinseriesexpansion,wecanwrite(1w)t=1Xi=0(t)iwi=i!; (m+1);r(6.4.42)=1Xi=0(t)i Usingthesubstitutionw=1ex,thesecondmomentofrthlgosfromtheGEDisgivenbyE(X(r;n;m;k))2=cr1

PAGE 83

ThejointpdfoftwolgosX(r;n;m;k)andX(s;n;m;k)(1r
PAGE 84

(m+1);sr: for1rr1isobtainedfromequations(6.4.40)and(6.4.46). ThemomentgeneratingfunctionoftwolgosisgivenbyMr;s;n;m;k(t1;t2)=Z102cs1et1x(1ex)(m+1)1 whereI(x)=Zx0et2y[(1ex)(m+1)(1ey)(m+1)]sr1(1ey)s1eydy: (m+1)(1ex)i;75

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(m+1);sr(1ex)r+1+i: (m+1);sr:Z10net1x(1ex)r+1(m+1)+i1:[1(1ex)(m+1)]r1exdxo:(6.4.48) Substitutingu=1exandusingtheexpansion(1u)t1=1Xj=0(t1)juj (m+1);srBr+i+j (m+1);r=1Xi=01Xj=0(t2)i(t1)jscsr1Y=01 Dierentiatingequation(6.4.49)withrespecttot1andthenwithrespecttot2andevaluatingatt1=t2=0,weobtaintheproductmomentsofX(r;n;m;k)andX(s;n;m;k)(1r
PAGE 90

and^=10V1(1010)V1h

PAGE 92

NotethatthecoecientsoftheBLUEfor,andthevariancecovariancearegivenrespectivelyinTable6.1,Table6.2andTable6.3form=0;k=1. Inparticular,if=1,fromTheorem6.5.1,wehave,d1=n+1;d2=(n+1)n;d3=(n+1)n(n1);:::,e1=n+2;e2=(n+2)(n+1);e3=(n+2)(n+1)n;:::,and,D0=(n+2)(n+1) n(n1)(n2)+:::+(n+2)(n+1)n:::3 n1; Form=1,k=1,j=1;j=1;2;:::,theBLUE^,^forandbasedonnlower83

PAGE 93

Inparticular,if=1,m=1,k=1,D0=9(3n11)=2,theBLUE^,^ofandfromtheuniformdistributionbasedonlowerrecordvaluesXL(1);XL(2);:::;XL(n)aregivenby^=1

PAGE 95

n(n+1)=(n+1)Xn;nX1;n

PAGE 96

+1s1:87

PAGE 97

xrxr+1:

PAGE 99

21-0.50000-1.000000-2.00000-2.50000-3.00000-3.50000-4.00000-5.00000221.500002.0000003.000003.500004.000004.500005.000006.0000041-0.07895-0.333333-1.09091-1.52439-1.97561-2.43839-2.90909-3.8659842-0.026320.0000000.181820.304880.439020.580570.727271.0309343-0.078950.0000000.272730.426830.585370.746450.909091.23711441.184211.3333331.636361.792681.951222.111372.272732.5979451-0.04412-0.250000-0.96000-1.38274-1.82707-2.28522-2.75269-3.7055352-0.011030.0000000.120000.207410.304510.408070.516130.7411153-0.025740.0000000.160000.262720.372180.48580.602150.8399254-0.077210.0000000.240000.367810.496240.62460.752691.00791551.158091.2500001.440001.544801.654141.766741.881722.116661-0.02727-0.200000-0.87591-1.29154-1.73159-2.18708-2.65285-3.6038362-0.005450.0000000.087590.154980.230880.312440.397930.5766163-0.010910.0000000.109490.185980.269360.357070.447670.6342764-0.025450.0000000.145990.235580.329220.425090.522280.7188465-0.076360.0000000.218980.329810.438950.546540.652850.86261661.145451.2000001.313871.385191.463181.545941.632121.8114971-0.01807-0.166667-0.81633-1.2268-1.66396-2.11779-2.58260-3.5327272-0.003010.0000000.068030.122680.184880.252120.322820.4710373-0.005420.0000000.081630.142310.209540.280930.355110.5087174-0.010840.0000000.102040.170770.244460.321060.399500.5595875-0.02530.0000000.136050.216310.298780.382220.466080.6341976-0.07590.0000000.204080.302830.398380.491430.582600.76103771.138551.1666671.224491.27191.327921.390031.456491.5981791-0.00915-0.12500-0.73587-1.13927-1.57279-2.0248-2.48874-3.4385292-0.001140.0000000.045990.085440.131070.180790.233320.3438593-0.001800.0000000.052560.095210.143550.195540.249990.363594-0.002990.0000000.061320.107900.15950.214170.270820.3877395-0.005390.0000000.073590.125170.180770.238640.297900.4187596-0.010780.0000000.091980.150200.210890.272730.335140.4606397-0.025150.0000000.122650.190260.257760.324680.390990.5220498-0.075460.0000000.183970.266360.343680.417450.488740.62645991.131861.1250001.103811.118711.145591.180791.221851.31555101-0.00685-0.11111-0.70697-1.10779-1.54012-1.99161-2.45541-3.40533102-0.000760.000000.039280.073850.114080.158060.204620.3027103-0.001140.000000.044190.081240.123590.169350.217410.31783104-0.001790.000000.05050.090520.135360.183180.232930.33599105-0.002990.000000.058910.102590.15040.200630.252350.35839106-0.005380.000000.0707000.119010.170450.223550.277580.38706107-0.010760.000000.088370.142810.198860.255490.312280.42577108-0.025110.000000.117830.180890.243050.304160.364320.48254109-0.075340.000000.176740.253250.324070.391060.455410.5790510101.130140.111111.060461.063631.080241.106131.138511.21600 Table6.1:CoecientsfortheBLUEof90

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213.000003.000003.7500004.200004.666675.142865.6256.6000022-3.00000-3.00000-3.750000-4.20000-4.66667-5.14286-5.625-6.60000411.618421.666672.352272.776833.223583.683994.153415.10928420.039470.00000-0.20455-0.33537-0.47561-0.62204-0.77273-1.08247430.118420.00000-0.30682-0.46951-0.63415-0.79976-0.96591-1.2989744-1.77632-1.66667-1.84091-1.97195-2.11382-2.26219-2.41477-2.72784511.461761.500002.156002.573363.015543.472943.940324.89375520.015440.00000-0.13200-0.224-0.32481-0.43139-0.54194-0.77075530.036030.00000-0.17600-0.28374-0.39699-0.51356-0.63226-0.87352540.108090.00000-0.26400-0.39723-0.52932-0.6603-0.79032-1.0482255-1.62132-1.50000-1.58400-1.66838-1.76441-1.86769-1.97581-2.20126611.369701.400002.032242.44432.883353.338853.805054.75729620.007270.00000-0.09489-0.16532-0.24371-0.32732-0.41451-0.59583630.014550.00000-0.11861-0.19838-0.28432-0.37408-0.46632-0.65542640.033940.00000-0.15815-0.25128-0.34751-0.44533-0.54404-0.74281650.101820.00000-0.23723-0.35179-0.46334-0.57257-0.68005-0.8913766-1.52727-1.40000-1.42336-1.47753-1.54447-1.61955-1.70013-1.87187711.308951.333331.946062.354042.790823.245053.710554.66223720.003870.00000-0.07289-0.12969-0.19369-0.26241-0.33435-0.48449730.006970.00000-0.08746-0.15044-0.21951-0.29240-0.36779-0.52325740.013940.00000-0.10933-0.18053-0.2561-0.33417-0.41376-0.57557750.032530.00000-0.14577-0.22867-0.31301-0.39782-0.48272-0.65231760.097590.00000-0.21866-0.32014-0.41735-0.51148-0.6034-0.7827877-1.46386-1.33333-1.31195-1.34458-1.39116-1.44677-1.50851-1.64383911.233401.250001.832312.234342.668083.120823.585654.53715920.001400.00000-0.04855-0.08924-0.13592-0.18652-0.2398-0.35149930.002200.00000-0.05548-0.09944-0.14887-0.20175-0.25693-0.37158940.003660.00000-0.06473-0.1127-0.16541-0.22097-0.27834-0.39635950.006590.00000-0.07768-0.13073-0.18746-0.24622-0.30617-0.42806960.013180.00000-0.09709-0.15688-0.2187-0.28139-0.34445-0.47086970.0307400.00000-0.12946-0.19871-0.2673-0.33499-0.40185-0.53365980.092230.00000-0.19419-0.2782-0.35641-0.4307-0.50232-0.6403799-1.38338-1.2500-1.16513-1.16844-1.18802-1.21827-1.25579-1.344791011.208221.222221.792322.192102.624793.077093.541794.493441020.000910.00000-0.04124-0.07681-0.11789-0.16258-0.20973-0.308751030.001370.00000-0.04640-0.08449-0.12771-0.17419-0.22284-0.324191040.002150.00000-0.05302-0.09414-0.13987-0.18841-0.23876-0.342711050.003590.00000-0.06186-0.10670-0.15541-0.20636-0.25865-0.365561060.006460.00000-0.07423-0.12377-0.17614-0.22994-0.28452-0.394811070.012920.00000-0.09279-0.14852-0.20549-0.26279-0.32008-0.434291080.030140.00000-0.12372-0.18813-0.25116-0.31285-0.37343-0.492191090.090410.00000-0.18558-0.26338-0.33488-0.40223-0.46679-0.590631010-1.35616-1.22222-1.11348-1.10618-1.11625-1.13774-1.16698-1.24032 Table6.2:CoecientsfortheBLUEof91

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20.066670.166670.333330.396830.450000.494950.533330.595240.600000.500000.562500.600000.633330.662340.687500.72857-0.13333-0.25000-0.41667-0.47619-0.52500-0.56566-0.60000-0.6547630.023810.075000.166670.201400.230110.254070.274290.306370.256610.200000.247690.272720.294190.312390.327860.35251-0.03968-0.100000-0.19444-0.22825-0.25568-0.27827-0.29714-0.3268040.011280.0444440.109090.133720.153940.170690.18470.206740.150380.1111110.150570.170130.186620.200410.211990.23020-0.01692-0.055556-0.12273-0.14709-0.16677-0.18288-0.19625-0.2170750.006130.0297620.080000.099340.115150.128180.139020.155960.100900.0714290.105130.121380.134940.146170.155550.17016-0.00858-0.035714-0.08800-0.10728-0.12283-0.13550-0.14598-0.1622060.003640.0214290.062570.078590.091670.102410.111310.125130.073130.0500000.079380.093340.104920.114460.122380.13466-0.00485-0.025000-0.06778-0.08383-0.10728-0.14598-0.11595-0.1293070.002300.0162040.051020.064760.075960.085130.092710.104430.055750.0370370.063030.075310.085440.093760.100640.11126-0.00296-0.018519-0.05466-0.06846-0.07958-0.08860-0.09602-0.1074180.001530.0126980.042850.054910.064730.072750.079370.089570.044050.0285710.0518500.062810.071840.079230.085330.0947-0.00191-0.014286-0.04553-0.05766-0.06743-0.07535-0.08185-0.0918190.001060.0102270.036790.047560.056320.063460.069350.078390.035760.0227270.043770.053690.061850.068500.073980.08238-0.00129-0.011364-0.03884-0.04967-0.05841-0.06548-0.07127-0.08013100.000750.0084180.032140.041870.049790.056240.061540.069670.029660.0185190.037700.046760.05420.060270.065250.072870.00090-0.009259-0.03374-0.04354-0.05145-0.05784-0.06308-0.07106 Table6.3:VariancesCovariancesoftheBLUEsofandintermsof292

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Thestudyofotherdistributionwithrespecttorecordsisnotverymuchexploited.Thiscouldbebecauseofthecomplexformoftheprobabilitydensityfunctionofrecordobservationsincecloseformsolutionswillnotbeobtainedforthemaximumlikelihoodestimates.How-ever,Themaximumlikelihoodestimatescanbeobtainedbyiterationandtheirpropertiesstudiedusingsimulations.Therefore,probabilitydensityfunctionsforexampletheJohnsonprobabilitydensityfunctioncanbeconsideredwithrespecttorecordbreakingdata.ThisisbecausetheJohnsonprobabilitydensityfunctionisveryusefulinlifescienceandengineering. Investigatingtheconceptofrecordswithrespecttousingthekerneldensityapproachtocharacterizethebehaviorofrecordsisaninterestingextensionofthetheoryofrecords.Thisapproachwillbeveryusefulincaseswhereaclassicaldistributioncannotbeidentiedtostatisticallyttheunderlyingdatafromwhichtherecordobservationsareobtained. Theinherentmissingdatastructurepresentintheseproblemsleadstolikelihoodfunc-tionsthatcontainpossiblyhigh-dimensionalintegrals,renderingtraditionalmaximumlikeli-hoodmethodsdicultornotfeasible.AninterestingextensionwillbetoobtainarbitrarilyaccurateapproximationstothelikelihoodfunctionbyiterativelyapplyingMonteCarloin-tegrationmethods(GeyerandThompson,[21]).SubiterationusingtheGibbssamplermayhelptoevaluateanymultivariateintegralsencounteredduringthisprocess. LetXn=(X(1)n;X(2)n);n=1;2;:::beindependentandidenticallydistributedR2randomvectorwithcommondistributionfunctionF.Toobtaintherecordsofsuchasequence,weemploythedenitionofthenaturalpartialorderingofR2,meaningXnisarecordifthereisarecordsimultaneouslyinbothcoordinates,thatis,X(1)n>X(1)j;j=1;:::;n1andX(2)n>X(2)j;j=1;:::;n1,seeGoldieandResnick,([23],[24])andGnedin,[22].An93

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Ahsanullah,M.(2005).Onlowergeneralizedorderstatisticsandacharacterizationofthepowerfunctiondistribution.StatisticalMethods.7(1),16-28.[2] Ahsanullah,M.(1995).RecordStatistics.NovaSciencePlublishersInc.NewYork,USA.[3] Ahsanullah,M.(2004).RecordStatistics-TheoryandApplications.UniversityPressofAmericaInc.USA.[4] Ahsanullah,M.(1996).Generalizedorderstatisticsfromtwoparameteruniformdistri-bution.Comm.Stat.Theor.Meth.25(10),2311-2318.[5] Ahsanullah,M.(1994).RecordsoftheGeneralizedExtremeValueDistribution.Pak.J.Statist,10(1)A,pp147-170.[6] Ahsanullah,M.andNevzorov,V.B.(2005).Orderstatistics:Examplesandexercises.NovaSciencePublishersInc.NewYork,USA.[7] Ahsanullah,M.andNevzorov,(2001).V.B."Orderrandomvariables,"NovaSciencePublishersInc.NewYork,USA.[8] Arnold,B.C.,Balakrishnan,N.andNagaraja,H.N.(1998).Records.JohnWiley,NewYork,NY,USA.[9] Balakrishnan,N.andChan,P.S.(1992).EstimationfortheScaleHalfLogisticsDis-tributionUnderTypeIICensoring.ComputationalStatistics&DataAnalysis(13)123-141.[10] Berkson,J.(1944)Applicationofthelogisticfunctiontobioassay,J.Amer.Statist.Assoc.39,357-365.[11] Berkson,J.(1951)WhyIpreferlogitstoprobits,Biometrics7,327-339.95

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Berkson,J.(1953)Astatisticallypreciseandrelativelysimplemethodofestimatingthebio-assayandquantalresponse,basedonthelogisticfunction,J.Amer.Statist.Assoc.48,565-599.[13] Burkschat,M.,Cramer,E.andKamps,U.(2003).DualGeneralizedOrderStatistics.Metron,LX1(1),13-26.[14] Calabria,R.andPulcini,G.(1990)Onthemaximumlikelihoodandleast-squaresesti-mationintheinverseWeibulldistribution,StatisticaApplicata,2,53-63.[15] Calabria,R.andPulcini,G.(1994)Bayes2-samplepredictionfortheinverseWeibulldistribution,Commun.Statist.-TheoryMeth.,23(6),1811-1824.[16] Carriere,J.(1992).ParametricModelsforLifeTables.TransactionsoftheSocietyofActuaries.44,77-100.[17] Chandler,K.M.(1952).ThedistributionandfrequencyofRecordValues.JournaloftheRoyalStatisticalSociety,ser.B14,220-228.[18] Coles,S.G.2001,AnIntroductiontoStatisticalModelingofExtremeValues,Springer,NewYork.[19] Feller,W.(1966).AnIntroductiontoProbabilityTheoryanditsApplications.Vol.II,Wiley,NewYork.[20] Galambos,J.(1987)TheAsymptoticTheoryTheoryofExtremeOrderStatistics,SecondEdition,Kreiger,Malabar,Florida.[21] GEYERC,.J.andTHOMPSONE,.A.(1992).ConstrainedMonteCarlomaximumlikelihoodfordependentdata(withdiscussion).J.R.Statist.Soc.B54,657-99.[22] GnedinA.V.(1993).ConialextremesofaMultivariateSample,Report,UniversitatGottinggen,InstitutfurMathematischeStochastik,Lotzestrasse13,3400Gottingen,Germany.[23] GoldieM.andResnick,S.I.(1995).OrderedIndependentScateringReport.SchoolofMath.Sci.,QueenMAryandWesteldcollege,MileEndRoad,London,E14NS,UK.[24] GoldieM.andResnick,S.I.(1989).RecordsofPartiallyOrderedact.Ann.Probab.17.678-689.96

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Gumbel,E.J.1958,StatisticsofExtremes,ColumbiaUniversityPress,NewYork.[26] Gulati,S.andPadgett,W.J.(2003).ParametricandNonparametricInferencefromRecord-BreakingData,NewYork:Springer.[27] Hosking,J.R.M.,Wallis,J.R.,andWood,E.F.1985,Estimationofthegeneralizedex-tremevaluedistributionbythemethodofprobabilityweightedmoments,Technometrics27,251-261.[28] Jaheen,Z.F.(2004).EmpiricalBayesinferenceforgeneralizedexponentialdistributionbasedonrecords.Comm.Stat.Ther.Meth.,33,1851-1861.[29] Kamps,U.(1995).Aconceptofgeneralizedorderedstatistics,B.G.TeubnerStuttgart,Germany.[30] Keller,A.Z.andKamath,A.R.R.(1982)Alternativereliabilitymodelsformechanicalsystems,ThirdInternationalConferenceonReliabilityandMaintainability,Toulose,France.[31] Kotz,S.2000,ExtremeValueDistributions:TheoryandApplications,ImperialCollegePress,London.[32] Lawless,J.F.(1982).StatisticalModels&MethodsForLifetimeData,JohnWiley&Sons.NewYork.[33] Llyod,E.H.(1952).LeastSquaresestimationoflocationandscaleparametersusingorderstatistics,Biometrika,39,88-95.[34] Luo,C.W.,Zhu,J.(2005)EstimatesoftheparametersoftheGumbeldistributionandtheirapplicationtoanalysisofwaterleveldata.(Chinese)ChineseJ.Appl.Probab.Statist.212,169-175.[35] Mbah,K.Alfred;Ahsanullah,M.(2007)"Somecharacterizationsofthepowerfunctiondistributionbasedonlowergeneralizedorderstatistics"PakistanJournalofStatistics,vol.23,pp.139-146.[36] Mbah,K.AlfredandAhsanullah,M.;EstimationofParametersofthePowerFunctionDistributionbyLowerGeneralizedOrderStatistics,underreview,JournalofStatisticalComputationandSimulation.97

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Mbah,K.AlfredandTsokos,P.Chris(2007)"OntheTheoryandApplicationofGumbelDistributionUsingRecords"FifthInternationalConferenceonDynamicSystemsandApplications,MorehouseCollege,Atlanta,Georgia,USA;May30-June2,20072.[38] Mbah,K.AlfredandTsokos,P.Chris"OntheTheoryofRecordsandApplications"Underreview,JournalofStatisticalTheoryandApplication.[39] McDonald,J.B.andXu,Y.J.(1995)Ageneralizationofthebetadistributionwithapplications.J.Econometrics66,133-152.[40] Mohamed,A.W.M,Khalaf,S.SandSafaa,M.A.(2003).OrderstatisticsfrominverseWeibulldistributionandcharacterizations.METRON,vol.LXI,n.3,pp.389-401[41] Nelson,W.(1982)AppliedLifeDataAnalysis,JohnWiley&Sons,NewYork.[42] Ojo,M.O.,(1989).AnalysisofSomePrisonData,J.Appl.Stat.16(1989),no6,377-383.[43] RaqabM.Z.(2002).Inferenceofgeneralizedexponentialdistributionbasedonrecordstatistics,J.Statist.Plann.Inference104,339-350.[44] RaqabM.Z.andAhsanullahM.(2001).Estimationofthelocationandscaleparametersofthegeneralizedexponentialdistributionbasedonorderstatistics,J.Statist.Comput.Simul.,69,109-123.[45] Roberts,E.M.(1979)ReviewofStatisticsofExtremevalueswithApplicationtoairqualitydata.PartII:application.J.AirPollutionControlAssoc.29,733-740.[46] Royston,J.P(1982)anextensionofShapiroandWilk'sWtestfornormalitytolargesamples,appl.Stat.31,115-124.[47] Samaniego,F.J.andWhitaker,L.R.(1986)OnestimatingPopulationCharacteristicsfromRecord-BreakingObservations.I.ParametricResults.NavalResearchLogisticsQuarterly,33,531-543[48] Sansalone,J.J.andBuchberger,S.G.(1997)"PartitioningandFirstFlushofMetalsinUrbanRoadwayStormWater."J.Env.Eng.123(2):134.[49] Weibull,W.(1951)"Astatisticaldistributionfunctionofwideapplicability"J.Appl.Mech.-Trans.ASME18(3),293-297.98

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DuringhisgraduatestudiesatUSF,AlfredMbahworkedasaTeachingassistantatthedepartmentofMathematicsandStatisticswherehetaughtseveralundergraduatecoursesinMathematicsandStatistics.