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Study of laplace and related probability distributions and their applications

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Study of laplace and related probability distributions and their applications
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Aryal, Gokarna Raj
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University of South Florida
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Subjects / Keywords:
Laplace distribution
Skewness
Truncation
Simulation
Reliability
Preventive maintenance
Renewal process
Dissertations, Academic -- Mathematics -- Doctoral -- USF
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bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

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Abstract:
ABSTRACT: The aim of the present study is to investigate a probability distribution that can be derived from the laplace probability distribution and can be used to model various real world problems. In the last few decades, there has been a growing interest in the construction of flexible parametric classes of probability distributions. Various forms of the skewed and kurtotic distributions have appeared in the literature for data analysis and modeling. In particular, various forms of the skew laplace distribution have been introduced and applied in several areas including medical science, environmental science, communications, economics, engineering and finance, among others. In the present study we will investigate the skew laplace distribution based on the definition of skewed distributions introduced by O'Hagan and extensively studied by Azzalini. A random variable X is said to have the skew-symmetric distribution if its probability density function is f(x) = 2g(x)G(lambda x), where g and G are the probability density function and the cumulative distribution function of a symmetric distribution around 0 respectively and lambda is the skewness parameter. We will investigate the mathematical properties of this distribution and apply it to real applications. In particular, we will consider the exchange rate data for six different currencies namely, Australian Dollar,Canadian Dollar, European Euro, Japanese Yen, Switzerland Franc and United Kingdom Pound versus United States Dollar. To describe a life phenomenon we will be mostly interested when the random variableis positive. Thus, we will consider the case when the skew Laplace pdf is truncated to the left at 0 and we will study its mathematical properties. Comparisons with other life time distributions will be presented. In particular we will compare the truncated skew laplace (TSL) distribution with the two parameter Gamma probability distribution with simulated and real data with respect to its reliability^ behavior. We also study the hypoexponential pdf and compare it with the TSL distribution. Since the TSL pdf has increasing failure rate (IFR) we will investigate a possible application in system maintenance. In particular we study the problem related to the preventive maintenance.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2006.
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Includes bibliographical references.
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by Gokarna Raj Aryal.
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Includes vita.

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ABSTRACT: The aim of the present study is to investigate a probability distribution that can be derived from the laplace probability distribution and can be used to model various real world problems. In the last few decades, there has been a growing interest in the construction of flexible parametric classes of probability distributions. Various forms of the skewed and kurtotic distributions have appeared in the literature for data analysis and modeling. In particular, various forms of the skew laplace distribution have been introduced and applied in several areas including medical science, environmental science, communications, economics, engineering and finance, among others. In the present study we will investigate the skew laplace distribution based on the definition of skewed distributions introduced by O'Hagan and extensively studied by Azzalini. A random variable X is said to have the skew-symmetric distribution if its probability density function is f(x) = 2g(x)G(lambda x), where g and G are the probability density function and the cumulative distribution function of a symmetric distribution around 0 respectively and lambda is the skewness parameter. We will investigate the mathematical properties of this distribution and apply it to real applications. In particular, we will consider the exchange rate data for six different currencies namely, Australian Dollar,Canadian Dollar, European Euro, Japanese Yen, Switzerland Franc and United Kingdom Pound versus United States Dollar. To describe a life phenomenon we will be mostly interested when the random variableis positive. Thus, we will consider the case when the skew Laplace pdf is truncated to the left at 0 and we will study its mathematical properties. Comparisons with other life time distributions will be presented. In particular we will compare the truncated skew laplace (TSL) distribution with the two parameter Gamma probability distribution with simulated and real data with respect to its reliability^ behavior. We also study the hypoexponential pdf and compare it with the TSL distribution. Since the TSL pdf has increasing failure rate (IFR) we will investigate a possible application in system maintenance. In particular we study the problem related to the preventive maintenance.
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StudyofLaplaceandRelatedProbabilityDistributionsand TheirApplications by GokarnaRajAryal Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:ChrisP.Tsokos,Ph.D. KandethodyRamachandran,Ph.D. GeoreyOkogbaa,Ph.D. GeorgeYanev,Ph.D. DateofApproval: May25,2006 Keywords:Laplacedistribution,Skewness,Truncation,Si mulation, Reliability,Preventivemaintenance,Renewalprocess. c r Copyright2006,GokarnaRajAryal

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Dedication ToMyParents

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Acknowledgements Iexpressmysincereanddeepestgratitudetomyresearchsup ervisoranddissertationadvisorProfessorC.P.Tsokos.Also,Iwouldliketop aymygratitudeto ProfessorA.N.V.Raoforhisguidancethroughoutmygraduat estudiesandprovidingvaluablesuggestionsinthisstudy.Thisworkcouldneve rhavebeencompleted withouttheirconstantguidanceandsupports. IwouldalsoliketothankDr.K.Ramachandran,Dr.G.Okogbaa andDr.G. Yanevfortheiradvice,supportandservinginmydissertati oncommittee. AspecialthankgoestoProfessorSureshKhatorforhiskindw illingnesstochair thedefenseofmydissertation. Lastbutnottheleast,Iamthankfultomyparents,mywifePra bhaandmy daughterPranshufortheirconstantsupportsandencourage ment.

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TableofContents ListofTables iv ListofFigures v Abstract vii 1Introduction 1 2TheLaplaceProbabilityDistribution 3 2.1Introduction................................32.2DenitionsandBasicProperties.................... .4 2.3DiscriminatingbetweentheNormalandLaplaceDistribu tions....7 2.4RepresentationandCharacterizations.............. ....8 2.5Conclusions................................11 3OntheSkewLaplaceProbabilityDistribution12 3.1Introduction................................123.2Moments.................................203.3MGFandCumulants...........................263.4Percentiles.................................283.5MeanDeviation..............................283.6Entropy..................................303.7Asymptotics................................323.8Estimation.................................333.9SimulationStudy.............................35 i

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3.10Conclusion.................................37 4ApplicationofSkewLaplaceProbabilityDistribution38 4.1Introduction................................384.2AnApplicationofSLDistributioninFinance.......... ...38 4.3Conclusion.................................45 5OntheTruncatedSkewLaplaceProbabilityDistribution46 5.1Introduction................................465.2Moments..................................485.3MGFandCumulants...........................515.4Percentiles.................................525.5MeanDeviation..............................525.6Entropy..................................535.7Estimation.................................555.8ReliabilityandHazardRateFunctions............... ..57 5.9MeanResidualLifeTimeandtheMeanTimeBetweenFailure ...60 5.10Conclusion.................................62 6ComparisonofTSLProbabilityDistributionwithOtherDis tributions63 6.1Introduction................................636.2TSLVs.TwoParameterGammaProbabilityDistribution.. ....63 6.3TSLVs.HypoexponentialProbabilityDistribution.... ......71 6.4Conclusion................................78 7PreventiveMaintenanceandtheTSLProbabilityDistribut ion80 7.1Introduction................................807.2AgeReplacementPolicyandTSLProbabilityDistributio n......81 7.3BlockReplacementPolicyandTSLProbabilityDistribut ion.....86 7.4MaintenanceOveraFiniteTimeSpan.................8 9 7.5Conclusion................................91ii

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8FutureResearch 92 References 95 AbouttheAuthor EndPageiii

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ListofTables 2.1SomepropertiesofNormal,LaplaceandCauchydistribut ions....6 2.2ParameterestimatorsofNormal,LaplaceandCauchydist ributions.7 3.1Valuesoftheparameter forselectedvaluesofskewness r ......24 4.1Descriptivestatisticsofthecurrencyexchangedata.. ........40 4.2EstimatedparametersandKolmogorov-SmirnovD-statis ticofcurrency exchangedata...............................40 4.3Kolmogorov-SmirnovD-statisticforcurrencyexchange datausingBoxCoxtransformations...........................41 6.1ComparisonbetweenTSLandgammamodelswhenbothparame ters areunknown................................66 6.2ComparisonbetweenTSLandgammamodelswhenoneparamet eris known...................................68 6.3TheReliabilityestimatesofPressureVesselsData.... ......70 6.4ComparisonbetweenTSLandHypoexponentialModels.... ....74 7.1Comparisonsofcostsfordierentvaluesofpreventivem aintenance times...................................91 iv

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ListofFigures 2.1PDFofstandardLaplace,NormalandCauchydistribution s.....5 3.1PDFofskewLaplacedistributionfor =1anddierentvaluesof .16 3.2Behaviorofexpectation,variance,skewnessandkurtos isofSL randomvariableasafunctionof for =1..............23 3.3P-PplotsofLaplaceandskewLaplacepdf'sfordierentv aluesof .36 4.1FittingSLmodelforAustralianDollarexchangeratadat a......42 4.2FittingSLmodelforCanadianDollarexchangeratedata. ......42 4.3FittingSLmodelforEuropeanEuroexchangeratedata... ....43 4.4FittingSLmodelforJapaneseYenexchangedata........ ...43 4.5FittingSLmodelforSwitzerlandFrancexchangeratedat a......44 4.6FittingSLmodelforUnitedKingdomPoundexchangerated ata...44 5.1PDFoftruncatedskewLaplacedistributionfor =1and =0 ; 1 ; 2 ; 5 ; 10 ; 50............................47 5.2Behaviorofexpectation,variance,skewnessandkurtos isof TSLdistribution.............................50 5.3ReliabilityandhazardrateofTSLdistributionfor =1.......59 6.1ReliabilityofTSLandGammadistributions........... ...65 6.2P-PPlotsofVesseldatausingTSLandGammadistribution .....69 6.3ReliabilityofVesseldatausingTSLandGammadistribut ion.....70 6.4PDFofHypoexponentialDistributionfor 1 =1anddierentvalues of 2 ....................................72 6.5ReliabilityandhazardratefunctionofHypoexponentia ldistribution.73 v

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6.6ReliabilityofTSLandHypoexponentialdistributionsf or n =50 ; 1 =1 ; and( a ) 2 =2 ; ( b ) 2 =5 ; ( c ) 2 =10 ; ( d ) 2 =20....75 6.7ReliabilityofTSLandHypoexponentialdistributionsf or n =100 ; 1 =1 ; and( a ) 2 =2 ; ( b ) 2 =5 ; ( c ) 2 =10 ; ( d ) 2 =20....76 6.8ReliabilityofTSLandHypoexponentialdistributionsf or n =500 ; 1 =1 ; and( a ) 2 =2 ; ( b ) 2 =5 ; ( c ) 2 =10 ; ( d ) 2 =20....77vi

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StudyofLaplaceandRelatedProbabilityDistributionsand TheirApplications GokarnaRajAryal ABSTRACT Theaimofthepresentstudyistoinvestigateaprobabilityd istributionthatcanbe derivedfromtheLaplaceprobabilitydistributionandcanb eusedtomodelvarious realworldproblems.Inthelastfewdecades,therehasbeena growinginterestinthe constructionofrexibleparametricclassesofprobability distributions.Variousforms oftheskewedandkurtoticdistributionshaveappearedinth eliteraturefordataanalysisandmodeling.Inparticular,variousformsoftheskewL aplacedistributionhave beenintroducedandappliedinseveralareasincludingmedi calscience,environmental science,communications,economics,engineeringandnan ce,amongothers.Inthe presentstudywewillinvestigatetheskewLaplacedistribu tionbasedonthedenition ofskeweddistributionsintroducedbyO'Haganandextensiv elystudiedbyAzzalini. Arandomvariable X issaidtohavetheskew-symmetricdistributionifitsproba bilitydensityfunctionis f ( x )=2 g ( x ) G ( x ),where g and G aretheprobabilitydensity functionandthecumulativedistributionfunctionofasymm etricdistributionaround 0respectivelyand istheskewnessparameter.Wewillinvestigatethemathemat ical propertiesofthisdistributionandapplyittorealapplica tions.Inparticular,wewill considertheexchangeratedataforsixdierentcurrencies namely,AustralianDollar,CanadianDollar,EuropeanEuro,JapaneseYen,Switzer landFrancandUnited KingdomPoundversusUnitedStatesDollar.Todescribealifephenomenonwewillbemostlyinterestedwh entherandomvariable ispositive.Thus,wewillconsiderthecasewhentheskewLap lacepdfistruncatedto theleftat0andwewillstudyitsmathematicalproperties.C omparisonswithother vii

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lifetimedistributionswillbepresented.Inparticularwe willcomparethetruncated skewlaplace(TSL)distributionwiththetwoparameterGamm aprobabilitydistributionwithsimulatedandrealdatawithrespecttoitsrelia bilitybehavior.Wealso studythehypoexponentialpdfandcompareitwiththeTSLdis tribution.Sincethe TSLpdfhasincreasingfailurerate(IFR)wewillinvestigat eapossibleapplicationin systemmaintenance.Inparticularwestudytheproblemrela tedtothepreventive maintenance.viii

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Chapter1 Introduction Thequalityoftheproceduresusedinastatisticalanalysis dependsheavilyonthe assumedprobabilitymodelordistributions.Becauseofthi s,considerableeortover theyearshasbeenexpendedinthedevelopmentoflargeclass esofstandarddistributionsalongwithrevelentstatisticalmethodologies,de signedtoserveasmodelsfor awiderangeofrealworldphenomena.However,therestillre mainmanyimportant problemswheretherealdatadoesnotfollowanyoftheclassi calorstandardmodels. Veryfewrealworldphenomenonthatweneedtostatistically studyaresymmetrical. Thusthepopularnormalmodelwouldnotbeausefulmodelfors tudyingeveryphenomenon.Thenormalmodelatatimesisapoordescriptionofo bservedphenomena. Skewedmodels,whichexhibitvaryingdegreesofasymmetry, areanecessarycomponentofthemodeler'stoolkit.Genton,M.[8]mentionsthata ctuallyanintroduction ofnon-normaldistributionscanbetracedbacktotheninete enthcentury.Edgeworth [7]examinedtheproblemofttingassymetricaldistributi onstoasymmetricalfrequencydata.Theaimofthepresentstudyistoinvestigateaprobabilityd istributionthatcanbe derivedfromtheLaplaceprobabilitydistributionandcanb eusedtomodelvarious realworldproblems.Infact,wewilldeveloptwoprobabilit ymodelsnamelytheskew LaplaceprobabilitydistributionandthetruncatedskewLa placeprobabilitydistributionandshowthatthesemodelsarebetterthantheexistingm odelstomodelsome oftherealworldproblems.Hereisanoutlineofthestudy:InchaptertwowewillstudythedevelopmentoftheLaplacepr obabilitydistribution 1

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anditsbasicproperties.Wewillmakeacomparisonsofthism odelwiththeGaussian distributionandtheCauchydistribution.Alsowewillpres entsomerepresentations oftheLaplacedistributionintermsofotherwellknowndist ributions. Inchapterthreewewillstudythestatisticalmodelcalledt heskewLaplaceprobability distribution.WiththetermskewLaplacewemeanaparametri cclassofprobability distributionsthatextendstheLaplaceprobabilitydistri butionbyadditionalshape parameterthatregulatesthedegreeofskewness,allowingf oracontinuousvariation fromLaplacetononLaplace.Wewillstudythemathematicalp ropertiesofthesubjectmodel.InchapterfourwewillpresentanapplicationoftheskewLap lacedistributionin nancialstudy.Infact,wewillusethecurrencyexchangeda taofsixdierentcurrencies,namely,AustralianDollar,CanadianDollar,Euro peanEuro,JapaneseYen, SwitzerlandFrancandtheUnitedKingdomPoundwithrespect totheUSDollar. Inchaptervewewilldevelopaprobabilitydistributionfr omtheskewLaplacedistributionpresentedinchaptertwo.Infact,wewilltruncat etheskewLaplacedistributionatzeroontheleftandwewillcallitthetruncatedske wLaplaceprobability distribution.Wewillpresentsomeofitsmathematicalprop erties. Inchaptersixwewillmakeacomparisonofthetruncatedskew Laplacedistribution withtwoexistingmodelsnamely,twoparametergammaandthe hypoexponential probabilitydistributions.Inchaptersevenwewillseekanapplicationofthetruncated skewLaplacedistributioninthemaintenancesystem.Wewilldevelopamodelthatc anbeusedtond theoptimumtimeinordertominimizethecostoveranitetim espan. Inthelastchapterwewillpresentpossibleextensionofthe presentstudy. 2

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Chapter2 TheLaplaceProbabilityDistribution 2.1IntroductionThequalityoftheproceduresusedinastatisticalanalysis dependsheavilyonthe assumedprobabilitymodelordistributions.Becauseofthi s,considerableeortover theyearshasbeenexpendedinthedevelopmentoflargeclass esofstandarddistributionsalongwithrevelentstatisticalmethodologies,de signedtoserveasmodelsfor awiderangeofrealworldphenomena.However,therestillre mainmanyimportant problemswheretherealdatadoesnotfollowanyoftheclassi calorstandardmodels. Theaimofthepresentstudyistoinvestigateaprobabilityd istributionthatcanbe derivedfromtheLaplacedistributionandcanbeusedonmode lingandanalyzing realworlddata.Inthe1923issueofthe JournalofAmericanStatisticalAssociation twopapersentitled"FirstandSecondLawsofError"byE.B.Wilsonand"Th euseofmedianin determiningseasonalvariation"byW.L.Crumwerepublishe d.IntherstpaperE.B WilsonstatesthatbothlawsoferrorwereoriginatedbyLapl ace. Therstlawproposedin1774,statesthatthefrequencyofan errorcouldbeexpressed asanexponentialfunctionofthenumericalmagnitudeofthe error,or,equivalently thatthelogarithmofthefrequencyofanerror(regardlesso fthesign)isalinear functionoftheerror.Thesecondlawproposedin1778,statesthatthefrequencyof theerrorisanexponentialfunctionofthesquareoftheerror,orequivalently thatthelogarithmofthe frequencyisaquadraticfunctionoftheerror. 3

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ThesecondLaplacelawiscalledthenormalorGaussianproba bilitydistribution. Sincetherstlawconsiststheabsolutevalueoftheerrorit bringsaconsiderable mathematicaldicultiesinmanipulation.Thereasonsfort hefargreaterattention beingpaidforthesecondlawisthemathematicalsimplicity becauseitinvolvesthe variable x 2 if x istheerror.TheLaplacedistributionisnamedafterPierre -Simon Laplace(1749-1827),whoobtainedthelikelihoodoftheLap lacedistributionismaximizedwhenthelocationparameterissettobethemedian.The Laplacedistribution isalsoknownasthelawofthedierencebetweentwoexponent ialrandomvariables.Consequently,itisalsoknownas doubleexponentialdistribution ,aswellasthe twotaileddistribution : Itisalsoknownasthe bilateralexponentiallaw 2.2DenitionsandBasicPropertiesThe classicalLaplaceprobabilitydistribution isdenotedby L ( ; )andisdenedby theprobabilitydensityfunction,pdf, f ( x ; ; )= 1 2 exp j x j ; 1 0arelocationandscaleparameters,respectively.This istheprobabilitydistributionwhoselikelihoodismaximi zedwhenthelocationparameteristobemedian.Itisasymmetricdistributionwhose tailsfallolesssharply thantheGaussiandistributionbutfasterthantheCauchydi stribution.Hence,itis ourinteresttocomparetheLaplacepdfwiththeGaussianpdf andCauchypdf.The probabilitydensityfunctionsoftheGaussianorNormal, N ( ; 2 )andtheCauchy, C ( x 0 ; )distributionsarerespectivelygivenby f ( x ; ; )= 1 p 2 exp ( x ) 2 2 2 (2.2.2) and f ( x ; x 0 ; )= 1 2 +( x x 0 ) 2 (2.2.3) 4

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where > 0, 1 << 1 > 0and 1
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TheLaplacepdfhasacusp,discontinuousrstderivative,a t x = ,thelocationparameter.Table2.1givessomeofthebasicandusefulpropert iesofLaplace,Gaussian andCauchypdf'sandtable2.2givesthecomparisonofthethe estimatesofsample mean,samplemedian,estimatorofsemi-interquartilerang e(S),whichisanestimator forhalf-widthathalfmaximum(HWHM)andthevarianceestim ator( s 2 )ofthese threepdf's.Thevarianceofthesamplemeanissimplythevarianceofthed istributiondivided bythesamplesizen.Forlargenthevarianceofthesamplemed ian m isgivenby V ( m )=1 = 4 nf 2 where f isthefunctionalvalueatthemedian. Bydenition S = 1 2 ( Q 3 Q 1 )and s 2 = 1 n 1 P ni =1 ( x i x ) 2 Hence, V ( S )= 1 4 ( V ( Q 1 )+ V ( Q 3 ) 2 Cov ( Q 1 ;Q 3 )) = 1 64 n 3 f 2 1 + 3 f 2 3 2 f 1 f 3 and V ( s 2 )= 4 22 n + 2 22 n ( n 1) where f 1 and f 3 arethefunctionalvaluesattherstquartile Q 1 andthethirdquartile Q 3 ; respectively.Also 2 and 4 arethesecondandfourthcentralmomentsofthe randomvariable. Distribution E ( X ) V ( X ) Sk(X) Kur(X) HWHM Char.Function Entropy Normal 2 0 3 p 2 ln 2 exp( it 2 t 2 2) ln( p 2 e ) Laplace 2 2 0 6 ln 2 1 1+ 2 t 2exp( it ) 1+ln(2 ) Cauchy Und: 1 Und: 1 exp( x0it j t j ) ln(4 ) Table2.1:SomepropertiesofNormal,LaplaceandCauchydis tributions6

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Distribution E ( X ) V ( X ) E(m) V(m) E ( s2) V ( s2) E(S) V(S) Normal 2 n 2 2 n 2 2 4 n 1 0 : 6745 1 16 nf ( Q 1 ) 2 Laplace 2 2 n 2 n 2 2 20 2 n ln 2 2 n Cauchy Und. 1 x0 2 2 4 n 1 1 2 2 4 n Table2.2:ParameterestimatorsofNormal,LaplaceandCauc hydistributionsWhere =1+ 0 : 4 n 1 ifweincludethesecondtermintheexpressionof V ( s 2 )and 1otherwiseandinthetableUnd.standsforundenedand m denotesthemedian. 2.3DiscriminatingbetweentheNormalandLaplaceDistribu tions BoththenormalandLaplacepdf'scanbeusedtoanalyzesymme tricdata.Itiswell knownthatthenormalpdfisusedtoanalyzesymmetricdatawi thshorttails,whereas theLaplacepdfisusedfordatawithlongtails.Although,th esetwodistributions mayprovidesimilardatatformoderatesamplesizes,howev er,itisstilldesirable tochoosethecorrectormorenearlycorrectmodel,sincethe inferencesofteninvolve tailprobabilities,andthusthepdfassumptionisveryimpo rtant. Foragivendataset,whetheritfollowsoneofthetwogivenpr obabilitydistribution functions,isaverywellknownandimportantproblem.Discr iminatingbetweenany twogeneralprobabilitydistributionfunctionswasstudie dbyCox[6]. RecentlyKundu[17]considerdierentaspectsofdiscrimin atingbetweentheNormalandLaplacepdf'susingtheratioofthemaximizedlikeli hoods(RML).Let X 1 ;X 2 ;:::X n bearandomsamplefromoneofthetwodistributions.Thelike lihood functions,assumingthatthedatafollow N ( ; 2 )or L ( ; ),are l N ( ; )= n Y i =1 f N ( X i ;; ) and l L ( ; )= n Y i =1 f L ( X i ;; ) ; 7

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respectively.ThelogarithmofRMLisdenedby T =ln ( l N ( b ; b ) l L ( b ; b ) ) : Notethat( b ; b )and( b ; b )arethemaximumlikelihoodestimatorsof( ; )and( ; ) respectivelybasedonarandomsample X 1 ;X 2 ;:::X n .Therefore,Tcanbewrittenas T = n 2 ln2 n 2 ln + n ln b n ln b + n 2 where b = 1 n n X i =1 X i ; b 2 = 1 n n X i =1 ( X i b ) 2 ; b =median f X 1 ;X 2 ;:::X n g ; b = 1 n n X i =1 j X i b j : Thediscriminationprocedureistochoosethenormalpdfift heteststatistic T> 0,otherwisechoosetheLaplacepdfasthepreferredmodel.N otethatifthenull distributionis N ( ; 2 ),thenthedistributionof T isindependentof and .Similarly ,ifthenulldistributionis L ( ; ),thenthedistributionof T isindependentof and 2.4RepresentationandCharacterizationsInthissectionwewouldliketopresentvariousrepresentat ionsofLaplacerandom variablesintermsoftheotherwellknownrandomvariablesa spresentedbyKotz et al. [15].ThesevariousformoftheLaplacepdfwillbeusefultot hepresentstudy. WeshallderivetherelationsforstandardclassicalLaplac eRandomvariablewhose probabilitydensityfunctionisgivenby f ( x )= 1 2 exp( j x j ) ; 1
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1.LetWbeastandardexponentialrandomvariable(r.v.)wit hprobabilitydensity f W ( w )=exp( w ) ;w> 0 andZbestandardnormalr.v.withprobabilitydensity f Z ( z )= 1 p 2 exp( z 2 = 2) ; 1 0 andZbestandardnormalr.v.withprobabilitydensity f Z ( z )= 1 p 2 exp( z 2 = 2) ; 1 0 andZbestandardnormaldistributionr.v.withprobability density f Z ( z )= 1 p 2 exp( z 2 = 2) ; 1
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5.Let Y 1 and Y 2 bei.i.d 2 r.vwithtwodegreesoffreedomi.e.havingtheprobability density f ( x )= 1 2 exp( x= 2) then X =( Y 1 Y 2 ) = 2hasstandardclassicalLaplacepdf. 6.LetWbeastandardexponentialr.vthen X = IW ,where I takesonvalues 1 withprobabilities1/2,hasstandardclassicalLaplacepdf 7.Let P 1 and P 2 arei.i.d.ParetoTypeIrandomvariableswithdensity f ( x )=1 =x 2 x 1then X =log( P 1 =P 2 )hasstandardclassicalLaplacepdf. 8.Let U 1 and U 2 bei.i.d.uniformlydistributedon[0 ; 1]then X =log( U 1 =U 2 ) hasstandardclassicalLaplacepdf.9.Let U i ;i =1 ; 2 ; 3 ; 4bei.i.d.standardnormalvariablesthenthedeterminant X = U 1 U 2 U 3 U 4 = U 1 U 4 U 2 U 3 hasstandardclassicalLaplacepdf.10.Let f X n ;n 1 g beasequenceofuncorrelatedrandomvariablesthen X = P 1n =1 b n X n hasaclassicalLaplacepdf,where, b n = n p 2 J 0 ( n ) Z 1 0 x exp( x ) J 0 ( n exp( x 2 )) dx and X n = p 2 n J 0 ( n ) J 0 ( n exp( j x j 2 )) ; where n isthenthrootof J 1 .Here, J 0 and J 1 aretheBesselfunctionsoftherst kindoforder0and1,respectively.TheBesselfunctionofth erstkindoforder i is 10

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denedby J i ( u )= u i 1 X k =0 ( 1) k u 2 k 2 2 k + k !( i + k +1) : ThisisalsocalledtheorthogonalrepresentationofLaplac erandomvariables. 2.5ConclusionsInthischapterwehavestudiedthedevelopmentofLaplacepr obabilitydistribution anditsbasicproperties.Morespecicallywehavederiveda nalyticalexpressionsfor allimportantstatisticsandtheircorrespondingestimate s,assummarizedinTable 2.1andTable2.2.Inadditionwehavedevelopedananalytica lcomparisonwiththe famousandhighlypopularGaussianprobabilitydistributi onandCauchyprobability distribution.Thereasonforthesubjectcomparisonisthef actthattheLaplacepdfis symmetricandwhosetailsfallolesssharplythantheGauss ianpdfbutfasterthan theCauchypdf.WealsoprovideamethodwhentochoosetheLaplacepdfoverGa ussianpdfin analyzingandmodelingarealworldphenomenon.Alistofvar iousrepresentationsof theLaplacepdfintermsofsomeotherwellknownandusefulpd f'sisalsoprovided. 11

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Chapter3 OntheSkewLaplaceProbabilityDistribution 3.1IntroductionVeryfewrealworldphenomenonthatweneedtostatistically studyaresymmetrical. Thusthepopularnormalmodelwouldnotbeausefulpdfforstu dyingeveryphenomenon.Thenormalmodelattimesisapoordescriptionofob servedphenomena. Skewedmodels,whichexhibitvaryingdegreesofasymmetry, areanecessarycomponentofthemodeler'stoolkit.Genton,M.[8]mentionsthata ctuallyanintroduction ofnon-normaldistributionscanbetracedbacktotheninete enthcentury.Edgeworth [7]examinedtheproblemofttingassymetricaldistributi onstoasymmetricalfrequencydata.OurinterestinthisstudyisabouttheskewLapl acepdf. WiththetermskewLaplace(SL)wemeanaparametricclassofp robabilitydistributionsthatextendstheLaplacepdfbyanadditionalshapepar ameterthatregulatesthe degreeofskewness,allowingforacontinuousvariationfro mLaplacetonon-Laplace. Ontheappliedside,theskewLaplacepdfasageneralization oftheLaplacelaw shouldbeanaturalchoiceinallpracticalsituationsinwhi chthereissomeskewness present. SeveralasymmetryformsofskewedLaplacepdfhaveappeared intheliterature. OneoftheearlieststudiesisduetoMcGill[19]whoconsider edthedistributionswith pdfgivenby f ( x )= 8><>: 1 2 exp( 1 j x j ) ; if x 2 2 exp( 2 j x j ) ; if x> (3.1.1) 12

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whileHolla etal. in1968studiedthedistributionwithpdfgivenby f ( x )= 8<: p exp( j x j ) ; if x (1 p ) exp( j x j ) ; if x> (3.1.2) where0 0. Therefore,if f isthestandardclassicalLaplacepdfgivenby f ( x )= 1 2 exp( j x j ) ; 1
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pdfgivenby f ( x )= 1 1+ 2 8>><>>: exp ( x ) ; if x exp 1 ( x ) ; if x< (3.1.5) where > 0and > 0.Notethatfor =1weobtainthepdfofsymmetricLaplace pdf.ThiswasintroducedbyHinkly etal. (1977)andthisdistributionistermedas asymmetricLaplace(AL) pdf.Anindepthstudyontheskew-Laplacedistribution wasreportedbyKotzetal.[14].Theyconsiderathreeparame tersskew-Laplace distributionwithpdfgivenby f ( x ; ;; )= 8><>: + exp( ( x )) ; if x + exp( ( x )) ; if x> (3.1.6) where isthemeanandtheparameters and describestheleftandright-tail shapes,respectively.Avalueof greaterthan suggeststhatthelefttailsare thinnerandthus,thatthereislesspopulationtotheleftsi deof thantotheright side;theoppositeisofcoursetrueif isgreaterthan .If = ,thedistributionis theclassicalsymmetricLaplacepdf.InthischapterwewillstudyindetailtheskewedLaplacepdf usingtheideaintroduced byO'HaganandextensivelystudiedbyAzzalini[4].ThestandardLaplacerandomvariablehastheprobabilityde nsityfunctionandthe cumulativedistributionfunction,cdf,speciedby g ( x )= 1 2 exp j x j (3.1.7) and G ( x )= 8>><>>: 1 2 exp x ; if x 0, 1 1 2 exp x ; if x 0, (3.1.8) 14

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respectively,where 1 0.Arandomvariable X issaidtohave theskewLaplacepdfwithskewnessparameter ,denotedbySL( ),ifitsprobability densityfunctionisgivenby f ( x )=2 g ( x ) G ( x ) ; (3.1.9) where x 2< and 2< ,therealline.TheLaplacepdfgivenby(3.1.7){(3.1.8)has beenquitecommonlyusedasanalternativetothenormalpdfi nrobustnessstudies; see,forexample,Andrews etal. [3]andHoaglin etal. [10].Ithasalsoattracted interestingapplicationsinthemodelingofdetectorrelat iveeciencies,extremewind speeds,measurementerrors,positionerrorsinnavigation ,stockreturn,theEarth's magneticeldandwindsheardata,amongothers.Themainfea tureoftheskewLaplacepdf(3.1.9)isthatanewparameter isintroducedtocontrolskewnessand kurtosis.Thus,(3.1.9)allowsforagreaterdegreeofrexib ilityandwecanexpectthis tobeusefulinmanymorepracticalsituations. Itfollowsfrom(3.1.9)thatthepdf f ( x )andthecdf F ( x )of X arerespectively givenby f ( x )= 8>><>>: 1 2 exp (1+ j j ) j x j ; if x 0, 1 exp j x j 1 1 2 exp x ; if x> 0 (3.1.10) and F ( x )= 8>><>>: 1 2 + sign( ) 2 1 1+ j j exp (1+ j j ) j x j 1 ; if x 0, 1 2 +sign( ) 1 2 exp j x j ( ) ; if x> 0, (3.1.11) where( )=1 1 2(1+ j j ) exp x Throughouttherestofourstudy,unlessotherwisestated,w eshallassumethat > 0 sincethecorrespondingresultsfor < 0canbeobtainedusingthefactthat X has apdfgivenby2 g ( x ) G ( x ). 15

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Figure3.1illustratestheshapeofthepdf(3.1.10)forvari ousvaluesof and =1. -3-2-10123 0.00.20.40.60.8 xf(x) -3-2-10123 0.00.20.40.60.8 -3-2-10123 0.00.20.40.60.8 -3-2-10123 0.00.20.40.60.8 -3-2-10123 0.00.20.40.60.8 -3-2-10123 0.00.20.40.60.8 -3-2-10123 0.00.20.40.60.8 -3-2-10123 0.00.20.40.60.8 -3-2-10123 0.00.20.40.60.8 -3-2-10123 0.00.20.40.60.8 -3-2-10123 0.00.20.40.60.8 -3-2-10123 0.00.20.40.60.8 -3-2-10123 0.00.20.40.60.8 -3-2-10123 0.00.20.40.60.8 -3-2-10123 0.00.20.40.60.8 -3-2-10123 0.00.20.40.60.8 -3-2-10123 0.00.20.40.60.8 -3-2-10123 0.00.20.40.60.8 l= 0 l= 1 l= 2 l= 10 l= 50 l=1 l=2 l=10 l=50 Probability density function of Skew Laplace distribution Figure3.1:PDFofskewLaplacedistributionfor =1anddierentvaluesof 16

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Thefollowingpropertiesareveryimmediatefromthedenit ion. Property1. ThepdfoftheSL(0)isidenticaltothepdfoftheLaplacepdf. Property2. As !1 ;f ( x ; )tendsto2 f ( x ) I x> 0 whichistheexponentialdistribution.Property3. If X hasSL( )then X hasSL( ) : OneinterestingsituationwheretheskewLaplacerandomvar iablemayoccuristhe following:Proposition .Let Y and W betwoindependent L (0 ; 1)randomvariablesand Z is denedtobeequalto Y conditionallyontheevent f Y>W g thentheresulting distribution Z willhaveskew-Laplacedistribution. Proof:Wehave P ( Z z )= P ( Y z j Y>W ) = P ( Y z;Y>W ) P ( Y>W ) = 1 P ( Y>W ) Z z 1 Z y 1 g ( y ) g ( w ) dwdy = 1 P ( Y>W ) Z z 1 g ( y ) G ( y ) dy Notethat P ( Y>W )= P ( Y W> 0)=1 = 2as Y W hasLaplacepdfwith mean0.Hence P ( Z z )=2 Z z 1 g ( y ) G ( y ) dy Dierentiatingtheaboveexpressionwithrespectto z weobtaintheskewLaplace pdf.Thispropositiongivesusaquiteecientmethodtogenerate randomnumbersfrom askewLaplacepdf.Itshowsthatinfactitissucienttogene rate Y and W from 17

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L(0,1)andset Z = 8<: Y; if Y>W Y; if Y W Again,ifweconsidertheLaplacedistributionwithlocatio nparameterbeing whichwecalltheclassicalLaplacedistribution(alsoknow nasrstlawofLaplace) denotedby CL ( ; )inthiscasethepdfandcdfofarerespectively g ( x )= 1 2 exp j x j and G ( x )= 8>><>>: 1 2 exp x ; if x 1 1 2 exp x ; if x Henceinthiscasefor > 0thecorrespondingpdfandcdfoftheskew-Laplace randomvariableare,respectively,givenby f ( x )= 8>><>>: 1 2 exp (1+ )( x ) ; if x 1 exp x 1 1 2 exp ( x ) ; if x> and F ( x )= 8>><>>: 1 2(1+ ) exp (1+ )( x ) ; if x 1 exp x 1 1 2 exp ( x ) if x 18

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TheskewLaplacepdf{inspiteofitssimplicity{appearsnot tohavebeenstudied indetail. Theonlyworkthatappearstogivesomedetailsofthisdistri butionisGupta et al. [9]wherethepdfofskewLaplacedistributionisgivenby f ( x )= exp( j x j = )[1+ sign ( x )(1 exp( j x j = ))] 2 x 2< (3.1.12) where 2< ,thereallineand > 0. Alsotheygivetheexpressionsfortheexpectation,varianc e,skewnessandthekurtosis.Buttheseexpressionsarenotentirelycorrectaspoint edoutbyAryal etal. [1].In thisstudywewillprovideacomprehensivedescriptionofth emathematicalpropertiesof(3.1.10)anditsapplications.Inparticular,wesha llderivetheformulasforthe k thmoment,variance,skewness,kurtosis,momentgeneratin gfunction,characteristic function,cumulantgeneratingfunction,the k thcumulant,meandeviationaboutthe mean,meandeviationaboutthemedian,Renyientropy,Shan non'sentropy,cumulativeresidualentropyandtheasymptoticdistributionofth eextremeorderstatistics. Weshallalsoobtaintheestimatesoftheseanalyticaldevel opmentsandperforma simulationstudytoillustratetheusefulnessoftheskew-L aplacedistribution.Our calculationsmakeuseofthefollowingspecialfunctions:t hegammafunctiondened by ( a )= Z 1 0 t a 1 exp( t ) dt ; thebetafunctiondenedby B ( a;b )= Z 1 0 t a 1 (1 t ) b 1 dt ; and,theincompletebetafunctiondenedby B x ( a;b )= Z x 0 t a 1 (1 t ) b 1 dt; (3.1.13) where a> 0, b> 0and0
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Wealsousearesultfromanalysiswhichsaysthatarootofthe transcendentalequation 1 x + wx =0(3.1.14) isgivenby x =1+ 1 X j =1 j j 1 w j j ;(3.1.15) see,forexample,page348inPolya etal. [22]. 3.2MomentsThemomentsofaprobabilitydistributionsisacollectiono fdescriptiveconstantsthat canbeusedformeasuringitsproperties.Usingthedenitio nofthegammafunction, itiseasytoshowthatthe k thmomentofaskewLaplacerandomvariable X isgiven by E X k = 8><>: k ( k +1) ; if k iseven, k ( k +1) 1 1 (1+ ) k +1 ; if k isodd. (3.2.16) Alsoweknowthat k X i =0 a k = 8>>>>><>>>>>: a 0 + k 2 X i =1 a 2 i + k 2 X i =1 a 2 i 1 ; if k iseven, a 0 + k 1 2 X i =1 a 2 i + k +1 2 X i =1 a 2 i 1 ; if k isodd. 20

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UsingtheBinomialexpansionand(3.2.16),the k thcentralmomentof X canbe derivedas E n ( X ) k o = 8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>: k + k 2 X j =1 k 2 j k 2 j 2 j (2 j +1) k 2 X j =1 k 2 j 1 k 2 j +1 2 j 1 (2 j )( )if k iseven k k 1 2 X j =1 k 2 j k 2 j 2 j (2 j +1) + k +1 2 X j =1 k 2 j 1 k 2 j +1 2 j 1 (2 j )( )if k isodd. (3.2.17) where =E( X )istheexpectationof X and( )=1 1 (1+ ) 2 j Itfollowsfrom(3.2.16)and(3.2.17)thattheexpectation, variance,skewnessand thekurtosisof X arederivedtobe Exp( X )= 1 1 (1+ ) 2 ; Var( X )= 2 2+8 +8 2 +4 3 + 4 (1+ ) 4 ; Ske( X )= 2 6+15 +20 2 +15 3 +6 4 + 5 2+8 +8 2 +4 3 + 4 3 = 2 ; andKur( X )= 3 8+64 +176 2 +272 3 +276 4 +192 5 +88 6 +24 7 +3 8 2+8 +8 2 +4 3 + 4 2 : Notethatthesefourexpressionsarevalidonlyfor > 0.ThecorrespondingexpressionsgiveninGupta etal. [9]arethesameastheabove,buttheyappeartoclaim thevalidityoftheexpressionsforall 2< 21

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AspointedoutbyAryal etal. [2]if < 0,onemustreplace by ineachofthe fourexpressions;inaddition,theexpressionsfortheexpe ctationandtheskewness mustbemultipliedby 1. Figure3.2illustratesthebehavioroftheabovefouranalyt icalexpressionsfor = 10 ;:::; 10.Boththeexpectationandtheskewnessareincreasingfun ctionsof with lim !1 E( X )= ; lim !1 Skewness( X )= 2 ; and lim !1 E( X )= ; lim !1 Skewness( X )=2 : Notethatthevarianceandthekurtosisareevenfunctionsof .Thevariancedecreases from2 2 to 2 as increasesfrom0to 1 whichisasignicantgainonintroducing thenewshapeparameter inthemodel.Thekurtosisdecreasesfor0 0 but thenincreasesforall > 0 ,where 0 isthesolutionoftheequationgivenby 4+6 =14 2 +56 3 +84 4 +70 5 +34 6 +9 7 + 8 : Numericalcalculationsshowthatfor 0 0 : 356.At =0, = 0 andas !1 thekurtosistakesthevalues6,5 : 810(approx)and9,respectively. 22

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-10-50510 -1.00.00.51.0 lE (X) -10-50510 1.01.41.8 lVariance(X) -10-50510 -2-1012 lSkewness(X) -10-50510 6.07.08.0 lKurtosis(X) Behavior of Expectation,Variance,Skewness and Kurtosis Figure3.2:Behaviorofexpectation,variance,skewnessan dkurtosisofSL randomvariableasafunctionof for =123

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WeknowthattheskewnessofarandomvariableXisdenedby r = thirdmomentaboutthemean (s.d) 3 Proposition :Thereisaonetoonecorrespondencebetween r ,theskewnessofa SLrandomvariableand ,theshapeparameterofSLpdf. Proof:WeknowthattheskewnessforaSkew-Laplacepdfisgiv enby r = 2 6+15 +20 2 +15 3 +6 4 + 5 2+8 +8 2 +4 3 + 4 3 = 2 ; whichcanbewrittenas r 2 = ( +1) 6 1 [( +1) 4 +2( +1) 2 1] 3 = 2 (3.2.18) andsetting( +1) 2 = x wehave r = 2( x 3 1) ( x 2 +2 x 1) 3 = 2 (3.2.19) Ifwearegivenavalueof r wecangetthecorrespondingvalueof x usingsimple calculations.Itisclearthat x ispositive.Table2.1belowgivessomevaluesof for agivenvaluesof r when r> 0 : r x 0 1 0 0 : 5 1 : 4319 0 : 1966 1 : 0 3 : 0409 0 : 7438 1 : 5 8 : 9808 1 : 9968 1 : 75 20 : 9875 3 : 5812 1 : 9 56 : 9946 6 : 5495 1 : 99999 599996 : 9989 773 : 5947 Table3.1:Valuesoftheparameter forselectedvaluesofskewness r24

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Notethatif r isgreaterthanorequalto2wewillhaveonlyimaginaryroots .Also notethatwhenwehave < 0 ; thenweknowthatthecorrespondingexpressionfor skewnessisobtainedonreplacing by andmultiplyingthewholeexpressionby -1. Inthiscasewehave r = 2( y 3 1) ( y 2 +2 y 1) 3 = 2 (3.2.20) where y =( 1) 2 : Againwecanndthevalueof y and onceweknowthevalueof r .Hence,knowingthevalueofskewnesswecancomputethecorr espondingunique valueof Considerthecasewhenthelocationparameterbeing and > 0the k thmoments aregivenbyE X k = 8>>>>>>>><>>>>>>>>: k exp( ) k +1; k 2 k +1 1 exp( 1 )( k +1; 1 ) + k 2 k +1 1 n exp( 1 )( k +1; 1 ) o if k iseven, k exp( ) k +1; + k 2 k +1 1 exp( 1 )( k +1; 1 ) k 2 k +1 1 n exp( 1 )( k +1; 1 ) o if k isodd. where 1 = +1 25

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3.3MGFandCumulantsThemomentgeneratingfunction,MGF,ofarandomvariable X isdenedby M ( t )= E (exp( tX )).When X hasthepdfgiven(3.1.10),directintegrationyieldsthat M ( t )= t ( t ) 2 (1+ ) 2 + 1 1 t for t< 1 = .Thus,thecharacteristicfunctiondenedby ( t )= E (exp( itX )andthe cumulantgeneratingfunctiondenedby K ( t )=log M ( t )areoftheform ( t )= it ( it ) 2 (1+ ) 2 + 1 1 it and K ( t )=log t ( t ) 2 (1+ ) 2 + 1 1 t ; respectively,where i = p 1.Byexpandingthecumulantgeneratingfunctionas K ( t )= 1 X k =1 a k ( t ) k k ; oneobtainsthecumulants a k givenby a k = 8>><>>: ( k 1)! k 1 1 (1+ ) 2 k ; if k isodd, ( k 1)! k 1+ 2 (1+ ) k 1 (1+ ) 2 k ; if k iseven. OneinterestingcharacterizationofaskewLaplacepdfisth efollowing: WehaveseenthatthecharacteristicfunctionofaSLrandomv ariableXisgivenby ( t )= it ( it ) 2 (1+ ) 2 + 1 1 it 26

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Itisclearthat X ( t )+ X ( t )= 1 1 it + 1 1+ it = 2 1 ( it ) 2 Infactwehavethefollowingproposition. Proposition :LetYbea L (0 ; 1)randomvariablewithprobabilitydensityfunction g ( x )andXbeSL( )derivedfromY,thentheevenmomentsofXareindependent of andarethesameasthatofY. Proof:Let X ( t )bethecharacteristicfunctionofXsothat X ( t )= Z 1 1 exp( itx )[2 g ( x ) G ( x )] dx: Now, X ( t )= Z 1 1 exp( itx )[2 g ( x ) G ( x )] dx = Z 1 1 exp( itz )[2 g ( z ) G ( z )] dz = Z 1 1 exp( itz )[2 g ( z )(1 G ( z ))] dz = Z 1 1 exp( itx )[2 g ( x )(1 G ( x ))] dx: Notethatthesecondfromthelastexpressionfollowsfromth efactthat g issymmetric about0.Hence,wehave h ( t )= X ( t )+ X ( t )=2 Z 1 1 exp( ity ) g ( y ) dy =2 Y ( t ) whichisindependentof : Alsowecanshowthatif( 1) n h (2 n ) (0) = 2and( 1) n (2 n ) Y (0)existthentheyarethe evenordermomentsofXandY,respectively,andtheyarethes ame. 27

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3.4PercentilesApercentileisameasureofrelativestandingofanobservat ionagainstallother observations.The pth percentilehasatleast p %ofthevaluesbelowthatpointand atleast(100 p )%ofthedatavaluesabovethatpoint.Toknowtheexpression of percentileisveryimportanttogeneraterandomnumbersfro magivendistribution. The100 p thpercentile x p isdenedby F ( x p )= p ,where F isgivenby(3.1.11).If 0 p F (0)=1 = f 2(1+ ) g theninverting F ( x p )= p ,onegetsthesimpleform x p = 1+ log f 2(1+ ) p g : (3.4.21) However,if1 = f 2(1+ ) g


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isanimportantdescriptivestatisticthatisnotfrequentl yencounteredinmathematicalstatistics.Thisisessentiallybecausewhileweconsi derthemeandeviationthe introductionoftheabsolutevaluemakesanalyticalcalcul ationsusingthisstatistic muchmorecomplicated.Butstillsometimesitisimportantt oknowtheanalytical expressionsofthesemeasures.Themeandeviationaboutthe meanandthemedian aredenedby 1 ( X )= Z 1 1 j x j f ( x ) dx and 2 ( X )= Z 1 1 j x M j f ( x ) dx; respectively,where = E ( X )and M denotesthemedian.Thesemeasurescanbe calculatedusingtherelationshipsthat 1 ( X )= Z 1 ( x ) f ( x ) dx + Z 1 ( x ) f ( x ) dx and 2 ( X )= Z 0 1 ( M x ) f ( x ) dx + Z M 0 ( M x ) f ( x ) dx + Z 1 M ( x M ) f ( x ) dx; where M> 0because F (0)=1 = f 2(1+ ) g < 1 = 2for > 0.Simplecalculationsyield thefollowingexpressions: 1 ( X )= 2 1 (1+ ) 2 exp 2 (2+ ) (1+ ) 2 exp (2+ ) (1+ ) 2 and 2 ( X )= M +2 exp M + (1+ ) 2 1 exp M (1+ ) + M 2(1+ ) exp M (1+ ) exp f (1+ ) g : 29

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Thecorrespondingexpressionsfor < 0arethesameasabovewith replacedby 3.6EntropyAnentropyofarandomvariable X isameasureofvariationoftheuncertainty.Renyi entropyisdenedby J R ( r )= 1 1 r log Z f r ( x ) dx ; (3.6.23) where r> 0and r 6 =1.See[24]fordetails.Forthepdf(3.1.10),notethat Z f r ( x ) dx = 1 (2 ) r Z 0 1 exp r (1+ ) x dx + 1 r Z 1 0 exp rx 1 1 2 exp x r dx: Bysubstituting y =(1 = 2)exp( x= )andthenusing(3.1.13),onecouldexpressthe aboveintermsoftheincompletebetafunction.Itfollowsth enthattheRenyientropy isgivenby J R ( r )= 1 1 r log 8<: + r (1+ )2 r (1+1 = ) B 1 = 2 r ;r +1 r 2 r r 1 (1+ ) 9=; : (3.6.24) Shannon'sentropydenedby E [ log f ( X )]isthelimitingcaseof(3.6.23)for r 1. InfactShannondevelopedtheconceptofentropytomeasuret heuncertaintyofa discreterandomvariable.SupposeXisadiscreterandomvar iablethatobtainsvalues fromaniteset x 1 ;x 2 ;:::;x n ,withprobabilities p 1 ;p 2 ;:::p n .Welookforameasure ofhowmuchchoiceisinvolvedintheselectionoftheeventor howcertainweareof theoutcome.Shannonarguedthatsuchameasure H ( p 1 ;p 2 ;:::p n )shouldobeythe followingproperties1.Hshouldbecontinuousin p i 2.Ifall p i areequalthenHshouldbemonotonicallyincreasingin n 3.Ifachoiceisbrokendownintotwosuccessivechoices,the originalHshouldbe 30

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theweightedsumoftheindividualvaluesofH.Shannonshowe dthattheonlyHthat satisesthesethreeassumptionsisoftheform H = k n X i =1 p i logp i andtermedittheentropyofX. Itiswellknownthatforanydistributionlimiting r 1inRenyiwegetthe shanonentropy.Hencefrom(3.6.24)andusingL'Hospital's ruleandtakingthelimits wehave E [ log f ( X )]=1+log(2 )+ 2(1+ ) 2 +2 1 = 1 1+ B 1 = 2 2+ 1 ; 0 B 1 = 2 1+ 1 ; 0 : Rao etal. [23]introducedthe cumulativeresidualentropy (CRE)denedby E ( X )= Z Pr( j X j >x )logPr( j X j >x ) dx; whichismoregeneralthanShannon'sentropyasthedenitio nisvalidinthecontinuousanddiscretedomains.However,theextensionofthisno tionofShanonentropy tocontinuousprobabilitydistributionpossessomechalla nges.Thestraightforward extensionofthediscretecasetocontinuousdistributionF withpdfdensity f called dierentialentropyandisgivenby H ( F )= Z f ( x )log f ( x ) dx: Howeverdierentialentropyhasseveraldrawbacksaspoint edoutinthepaperby Rao etal. [23]. Forthepdf(3.1.10),notethat Pr( j X j >x )=exp x 31

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as approaches 1 .Thus,inthiscasewehave logPr( j X j >x )= x for x> 0.Hence,theCREtakesthesimpleform, E ( X )= 3.7AsymptoticsIf X 1 ;:::;X n isarandomsamplefrom(3.1.10)andif X =( X 1 + + X n ) =n denotes thesamplemeanthenbyusingtheCentralLimitTheorem, p n ( X E ( X )) = p Var ( X ) approachesthestandardnormaldistributionas n !1 .Sometimesonewouldbe interestedintheasymptoticsoftheextremevalues, M n =max( X 1 ;:::;X n )and m n =min( X 1 ;:::;X n ).Forthecdf(3.1.11),itcanbeseenthat lim t !1 1 F ( t + x ) 1 F ( t ) =exp( x ) and lim t !1 F t 1+ x F ( t ) =exp( x ) : Thus,itfollowsfromTheorem1.6.2inLeadbetter etal. [15]thattheremustbe normingconstants a n > 0, b n c n > 0and d n suchthat Pr f a n ( M n b n ) x g! exp f exp( x ) g and Pr f c n ( m n d n ) x g! 1 exp f exp( x ) g as n !1 .Theformofthenormingconstantscanalsobedetermined.Fo rinstance, usingCorollary1.6.3inLeadbetter etal. [18],onecanseethat a n =1 = and b n = log n 32

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3.8EstimationGivenarandomsample X 1 ;X 2 ;:::;X n from(3.1.10),wewishtoestimatetheparametersstatedabovebyusingthemethodofmoments.Byequa tingthetheoretical expressionsfor E ( X )and E ( X 2 )withthecorrespondingsampleestimates,oneobtainstheequations: 1 1 (1+ ) 2 = m 1 (if > 0) ; (3.8.25) 1 (1 ) 2 1 = m 1 (if < 0)(3.8.26) and 2 2 = m 2 ; (3.8.27) where m 1 = 1 n n X i =1 x i and m 2 = 1 n n X i =1 x 2i : From(3.8.27),oneobtainanestimateoftheparameter ,givenby b = r m 2 2 : (3.8.28) Substitutingthisinto(3.8.25)and(3.8.26),onegetthees timateof ,namely b = 1 m 1 r 2 m 2 1 = 2 1(3.8.29) 33

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and b =1 1+ m 1 r 2 m 2 1 = 2 (3.8.30) respectively.Notethat b in(3.8.29)ispositiveifandonlyif m 1 > 0and b in(3.8.30) isnegativeifandonlyif m 1 < 0.Thus,dependingonwhether m 1 > 0or m 1 < 0, onewouldchooseeither(3.8.29)or(3.8.30)astheestimate of TheestimationoftheparametersbythemethodofMaximumlik elihood,MLE,is describedbelow.Let X 1 ;X 2 ;::::;X n bearandomsamplefromtheskewLaplacepdf. Thenthelikelihoodfunctionisgivenby L ( ; ; x 1 ;x 2 ;:::;x n )= j Y i =1 1 2 exp( (1+ ) x i ) n Y i = j +1 1 exp( x i )[1 1 2 exp( x i )] whereweassumethattherst j observationstakenegativevaluesandtherestassume thepositivevalues.Toestimatetheparameters and weconsiderthelog-likelihood andsetequaltozeroafterdierentiatingwithrespectto and respectivelyandwe getthefollowingpairofequations j X i =1 x i + n X i = j +1 x i exp( x i ) 2 exp( x i ) =0(3.8.31) and n + 1 j X i =1 x i 1 n X i = j +1 x i =0 ; (3.8.32) SolvingthissystemofequationsweobtaintheMLEof b = 1 n n X i = j +1 x i j X i =1 x i # (3.8.33) 34

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Now,toestimate onecansimplysubstitutethevalueof b in(3.8.31).Infactif wesupposethatalltheobservationsinarandomsampleiscom ingfromarandom variable X SL ( ; ),with > 0thenthemaximumlikelihoodmethodproduces b = X and b = 1 ,thatis,itisasthoughweareestimatingthedatasetascomi ng fromanexponentialdistribution.3.9SimulationStudyInthissectionweperformasimulationstudytoillustratet herexibilityof(3.1.10) over(3.1.7).Anidealtechniqueforsimulatingfrom(3.1.1 0)istheinversionmethod. (i)If > 0then,usingequations(3.4.21)and(3.4.22),onewouldsim ulate X by X = 1+ log f 2(1+ ) U g (3.9.34) if0 U 1 = f 2(1+ ) g andby X = log ( 1 U +(1 U ) 1 X j =1 (1+ ) j j 1 (1 U ) j j 2 j (1+ ) j ) (3.9.35) if1 = f 2(1+ ) g 0then,usingequationsonewouldsimulate X by X = 8>>>>>><>>>>>>: 1+ log f 2(1+ ) U g ; if0 U 1 = f 2(1+ ) g log ( 1 U +(1 U ) 1 X j =1 (1+ ) j j 1 (1 U ) j j 2 j (1+ ) j ) ; if1 = f 2(1+ ) g
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positivelyskewedwhiletheotherisnegativelyskewed.We ttedboththesesamples tothetwomodelsdescribedbythestandardLaplacepdf(equa tion(3.1.7))andthe skewLaplacepdf(equation(3.1.10)).Weusedthemethodofm omentsdescribedby equations(3.8.28){(3.8.30)toperformthetting.Allthe necessarycalculationswere implementedbyusingthe R languagebyIhaka et.al, [11].TheP-Pplotsarisingfrom thesetsareshowninthefollowingFigure3.3.Itisevidentthattheskew-Laplacedistributionprovidesa verysignicantimprovementoverstandardLaplacepdfforbothpositivelyandnegat ivelyskeweddata. 0.00.20.40.60.81.0 0.00.20.40.60.81.0 LaplaceObservedExpected 0.00.20.40.60.81.0 0.00.20.40.60.81.0 Skew LaplaceObservedExpected p-p plots for phi=1,lambda=1 0.00.20.40.60.81.0 0.00.20.40.60.81.0 LaplaceObservedExpected 0.00.20.40.60.81.0 0.00.20.40.60.81.0 Skew LaplaceObservedExpected pp plot for phi=1 and lambda=-2 Figure3.3:P-PplotsofLaplaceandskewLaplacepdf'sfordi erentvaluesof 36

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3.10ConclusionInthischapterwehavecompletelydevelopedtheskewLaplac eprobabilitydistribution.Thatis,itsmathematicalproperties,analyticalexp ressionsforallimportant statisticalcharacterizationalongwiththeestimations. Utilizingaprecisenumerical simulationwehaveillustratedtheusefulnessofouranalyt icaldevelopmentswithrespecttopositiveandnegativeaspectsoftheskewLaplacepd f. Finally,wehaveconcludedthatthesubjectpdfisbettermod elforskewedtypeof datathantheotherpopularmodels.Itiseasiertoworkwithb ecauseofitsanalytical tractabilityandtheonetoonecorrespondencebetweenthes kewness r andtheshape parameter ofthemodel. 37

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Chapter4 ApplicationofSkewLaplaceProbabilityDistribution 4.1IntroductionVariousversionsoftheLaplaceandskewLaplacepdfhavebee nappliedinsciences, engineeringandbusinessstudies.RecentlyJulia et.al [12]hasappliedtheskew LaplacepdfinGram-negativebacterialaxeniccultures.In thisstudythecytometric sidelightscatter(SS)valuesinGram-negativebacteriawe rettedusingtheskewLaplacepdfproposedbyKotz. et.al [15]givenby f ( x ; ;; )= 8><>: + exp( ( x )) ; if x + exp( ( x )) ; if x> 4.2AnApplicationofSLDistributioninFinanceInthepresentstudywewillpresentanapplicationoftheske wLaplacemodelpresentedinthepreviouschapterformodelingsomenancialda ta.Actuallyanarea wheretheLaplaceandrelatedprobabilitydistributionsca nndmostinterestingand successfulapplicationisonmodelingofnancialdata.Tra ditionallythesetypeof dataweremodeledusingtheGaussianpdfbutbecauseoflongt ailsandasymmetry presentinthedataitisnecessarytolookforaprobabilityd istributionwhichcan accountfortheskewnessandkurtosisdieringfromGaussia n.SincetheLaplacepdf canaccountforleptokurticbehavioritisthenaturalchoic eandmoreoverifskewness ispresentinthedatathentheskewLaplacepdfwilltakecare ofit.Hencetheskew 38

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Laplacepdfshouldbeconsideredastherstchoiceforskewe dandkurtoticdata. Klein[14]studiedyieldinterestratesonaveragedaily30y earTreasurybondfrom 1977to1990andfoundthatempiricaldistributionistoopea kyandfat-tailedsothe normalpdfwon'tbeanappropriatemodel.Kozubowski etal. [16]suggestedthatan asymmetricLaplacemodeltobetheappropriatemodelforint erestratedataarguingthatthismodeliseasyandcapableofcapturingthepeake dness,fat-tailedness, skewnessandhighkurtosispresentinthedata.Actuallythe yttedthemodelfor thedatasetconsistingofinterestrateson30yearTreasury bondsonthelastworking dayofthemonthcoveringtheperiodofFebruary1977through December1993. KozubowskiandPodgorski[16]ttedasymmetricLaplace(AL )modelwhosedensity isgivenby f ( x;; )= 1 1+ 2 exp x + 1 x ;x 2R ;> 0 ;> 0 where x + =max f x; 0 g & x =max f x; 0 g totthedatasetoncurrencyexchange. ActuallytheyttedthemodelforGermanDeutschmarkvs.U.S .Dollarandthe JapaneseYenVs.theU.S.Dollar.Theobservationweredaily exchangeratefrom 01/01/1980t012/07/1990,approximately2853datapoints. Here,weshallillustrateanapplicationoftheskewLaplace pdf(3.1.10)thatwe havestudiedinthepreviouschaptertothenancialdata.Th edataweconsiderare annualexchangeratesforsixdierentcurrenciesascompar edtotheUnitedStates Dollar,namely,AustralianDollar,CanadianDollar,Europ eanEuro,JapaneseYen, SwitzerlandFrancandUnitedKingdomPound.Thedatawereob tainedfromthe website http://www.globalndata.com/ .Thestandardchangeinthelog(rate)from yearttoyeart+1isused.TheskewLaplacepdfwasttedtoeachofthesedatasetsbyusi ngthemethodof maximumlikelihood.Aquasi-Newtonalgorithmin R wasusedtosolvethelikelihood equations.Table4.1showstherangeofthedataandthedescr iptivestatisticsofthe data.Itincludesnumberofobservations(n),mean,standar ddeviation(SD),Skewness (SKEW)andkurtosis(KURT)ofthedata.Estimationofthepar ameters and 39

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andtheKolmogorov-SmirnovDstatisticconsideringtheNor mal,theLaplaceand theSLmodelsisgiveninTable4.2forthesubjectdata. Currency YearsofData n Mean SD SKEW KURT AustralianDollar 1822-2003 182 3 : 626 1 : 648 0 : 703 2 : 047 CanadianDollar 1858-2003 146 1.106 0.176 2.882 16.12 EuropeanEuro 1950-2003 54 1.088 0.147 0.307 3.06 JapaneseYen 1862-2003 142 101.672 138.918 0.981 2.294 SwitzerlandFranc 1819-2003 185 4.105 1.269 -0.932 2.845 UnitedKingdomPound 1800-2003 204 4.117 1.384 0.0264 5.369 Table4.1:Descriptivestatisticsofthecurrencyexchange data Currency b b DNormal DLaplace DSkewLaplace AustralianDollar 0 : 0431 0 : 0123 0 : 3414 0 : 2983 0 : 1823 CanadianDollar 0 : 3846 0 : 0182 0 : 4125 0 : 3846 0 : 1254 EuropeanEuro 0 : 5556 0 : 0157 0 : 2310 0 : 1765 0 : 1471 JapaneseYen 0 : 0651 0 : 0131 0 : 3140 0 : 2971 0 : 2246 SwitzerlandFranc 0 : 0312 1 : 627 e 07 0 : 3168 0 : 2772 0 : 1957 UnitedKingdomPound 0 : 0245 0 : 0039 0 : 3114 0 : 2709 0 : 1478 Table4.2:EstimatedparametersandKolmogorov-SmirnovDstatisticofcurrencyexchange dataThegures(4.1-4.6)showhowwellthethisnancialdatats theskewLaplacepdffor thesubjectdatasets.Itisevidentthatthetsaregood.The Kolmogorov-Smirnov D-statisticonttingtheSLpdfiscomparedwiththeNormala ndLaplacepdf'sfor eachdata.Table4.2showsthatforeachdatatheskewLaplace pdftsbetterthan thetheGaussianandtheLaplacepdf.WealsottedthedataconsideringtheBox-coxtransformati on.Thisisatransformationdenedby f ( x )= 8<: x 1 if 6 =0 log x if =0 : 40

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UsingtheBox-Coxtransformationwehaveasignicantimpro vementoverthelog transformation.Table4.3showsthevaluesofthetransformationparameter .AlsothetableincludestheestimatedparametersoftheskewLaplacepdffort ransformeddataand theKolmogorov-SmirnovD-statisticfortheSLpdf,theLapl acepdfandtheNormal pdf. Currency b b b DNormal DLaplace DSkewLaplace AustralianDollar 1 : 36824 0 : 053 0 : 0137 0 : 3133 0 : 2486 0 : 1326 CanadianDollar 5 : 00 0 : 357 0 : 0162 0 : 4040 0 : 3675 0 : 0461 EuropeanEuro 0 : 5 0 : 555 0 : 0181 0 : 2330 0 : 1765 0 : 1471 JapaneseYen 0 : 1727 0 : 0630 0 : 0098 0 : 3349 0 : 3043 0 : 2536 SwitzerlandFranc 2 0 : 0530 0 : 0094 0 : 3085 0 : 2446 0 : 1413 UnitedKingdomPound 1 0 : 0369 0 : 0069 0 : 2950 0 : 2414 0 : 1281 Table4.3:Kolmogorov-SmirnovD-statisticforcurrencyex changedatausingBox-CoxtransformationsIneachofthegures(4.1-4.6)wecanseethattheactualdata ispickyandskewed soitisobviousthatneithertheusualGaussiannortheLapla cepdfwouldtthedata. Hence,wechooseSLpdftotthedatausingbothlogtransform ationandBox-Cox transformation.Infact,ifwecarefullyobservethegures thedatawithBox-Cox transformationsupporttheSLpdfbetterthanthelogtransf ormation. 41

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PercentageChangeinExchangeRatesofAustralianDollarto USDollar -101234 024681012 -101234 024681012 Empirical PDFFitted Skew Laplace PDF Percentage Change in Exchange Rate/100 (log transformation)PDF -101234 02468 -101234 02468 Empirical PDFFitted Skew Laplace PDF Percentage Change in Exchange Rate/100 (Box-Cox transformation)PDF Figure4.1:FittingSLmodelforAustralianDollarexchange ratadata PercentageChangeinExchangeRatesofCanadianDollartoUS Dollar 020406080 0.00.20.40.60.81.01.2 020406080 0.00.20.40.60.81.01.2 Percentage Change in Exchange Rate/100 (log transformation)PDF 0204060 0.00.20.40.60.81.01.21.4 0204060 0.00.20.40.60.81.01.21.4 Empirical PDFFitted Skew Laplace PDF Percentage Change in Exchange Rate/100 (Box-Cox transformation)PDF Figure4.2:FittingSLmodelforCanadianDollarexchangera tedata42

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PercentageChangeinExchangeRatesofEuropeanEurotoUSDo llar -10-8-6-4-2024 0.00.20.40.60.8 -10-8-6-4-2024 0.00.20.40.60.8 Percentage Change in Exchange Rate/100 (log transformation)PDF -10-8-6-4-2024 0.00.20.40.60.8 -10-8-6-4-2024 0.00.20.40.60.8 Empirical PDFFitted Skew Laplace PDF Percentage Change in Exchange Rate/100 (Box-Cox transformation)PDF Figure4.3:FittingSLmodelforEuropeanEuroexchangerate data PercentageChangeinExchangeRatesofJapaneseYentoUSDol lar -3-2-1012 0246 -3-2-1012 0246 Empirical PDFFitted Skew Laplace PDF Percentage Change in Exchange Rate/100 (log transformation)PDF -3-2-1012 02468 -3-2-1012 02468 Empirical PDFFitted Skew Laplace PDF Percentage Change in Exchange Rate/100 (Box-Cox transformation)PDF Figure4.4:FittingSLmodelforJapaneseYenexchangedata43

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PercentageChangeinExchangeRatesofSwitzerlandFrancto USDollar -0.50.00.51.0 02468101214 -0.50.00.51.0 02468101214 Empirical PDFFitted Skew Laplace PDF Percentage Change in Exchange Rate/100 (log transformation)PDF -0.50.00.51.01.5 02468 -0.50.00.51.01.5 02468 Empirical PDFFitted Skew Laplace PDF Percentage Change in Exchange Rate/100 (Box-Cox transformation)PDF Figure4.5:FittingSLmodelforSwitzerlandFrancexchange ratedata PercentageChangeinExchangeRatesofUnitedKingdomPound toUSDollar -0.50.00.51.01.5 051015 -0.50.00.51.01.5 051015 Empirical PDFFitted Skew Laplace PDF Percentage Change in Exchange Rate/100 (log transformation)PDF -0.50.00.51.01.5 024681012 -0.50.00.51.01.5 024681012 Empirical PDFFitted Skew Laplace PDF Percentage Change in Exchange Rate/100 (Box-Cox transformation)PDF Figure4.6:FittingSLmodelforUnitedKingdomPoundexchan geratedata44

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4.3ConclusionInthepresentstudywehaveidentiedarealworldnanciald ata,thatis,theexchange rateforsixdierentcurrencies,namely,AustralianDolla r,CanadianDollar,European Euro,JapaneseYen,SwitzerlandFrancandUnitedKingdomPo undwithrespectto USDollar.Traditionallythenancialanalystsareusingth eclassicalGaussianpdfto modelsuchdata.WehaveshownthattheSLpdftssignicantl ybetterthesubject datathantheGaussianandtheLaplacepdf.Thus,inperformi nginferentialanalysis ontheexchangeratedata,onecanobtainmuchbetterresults whichwillleadto minimizingtheriskinadecisionmakingprocess.Thegoodne ssoftcomparisons wasbasedontwodierentapproach,namely,usingthelogtra nsformationandusing theBox-Coxtransformation.Table4.2andtable4.3showtha tineithercasetheSL pdftsbetterthesubjectdatathantheGaussianpdfandLapl acepdf. 45

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Chapter5 OntheTruncatedSkewLaplaceProbabilityDistribution 5.1IntroductionTodescribealifephenomenonwewillbemostlyinterestedwh entherandomvariable ispositive.Thus,wenowconsiderthecasewhenskewLaplace pdfistruncatedtothe left0.Throughoutthisstudy,unlessotherwisestatedwesh allassumethat > 0.In thiscasewecanwrite F ( x )=Pr( X 0)= R x 0 f ( t ) dt 1 F (0) = F ( x ) F (0) 1 F (0) Hence,itcanbeshownthatthecdfofthetruncatedSkewLapla ce,TSL,random variableisgivenby F ( x )=1+ exp (1+ ) x 2(1+ )exp x (2 +1) (5.1.1) andthecorrespondingprobabilitydensityfunctionby f ( x )= 8><>: (1+ ) (2 +1) 2exp x exp (1+ ) x if x> 0, 0 otherwise (5.1.2) Aryal etal. [1],proposedthisprobabilitydistributionasareliabili tymodel. 46

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Agraphicalpresentationof f ( x )for =1andvariousvaluesof isgivenin Figure5.1. 0246810 0.00.20.40.60.81.0 xf(x) 0246810 0.00.20.40.60.81.0 0246810 0.00.20.40.60.81.0 0246810 0.00.20.40.60.81.0 0246810 0.00.20.40.60.81.0 l= 0 l= 1 l= 5 l= 10 l= 50 PDF of Truncated Skew-Laplace Distribution Figure5.1:PDFoftruncatedskewLaplacedistributionfor =1and =0 ; 1 ; 2 ; 5 ; 10 ; 5047

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Inthisstudywewillprovideacomprehensivedescriptionof themathematical propertiesof(5.1.2).Inparticular,weshallderivethefo rmulasforthe k thmoment, variance,skewness,kurtosis,momentgeneratingfunction ,characteristicfunction,cumulantgeneratingfunction,the k thcumulant,meandeviationaboutthemean,expressionsforRenyientropy,Shannon'sentropy,cumulati veresidualentropy.Alsowe willstudyreliabilityandhazardratebehaviorofthesubje ctpdf. 5.2MomentsIf X bearandomvariablewithpdfgivenby(5.1.2),thenusingthe denitionofthe gammafunction,itiseasytoshowthatthe k thmomentof X isgivenby E X k = k (1+ )( k +1) (2 +1) 2 1 (1+ ) k +1 : (5.2.3) UsingtheBinomialexpansionand(5.2.3),the k thcentralmomentof X canbederived tobegivenby E n ( X ) k o = 8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>: k + k 2 X j =1 k 2 j k 2 j 2 j (1+ )(2 j +1) (2 +1) n 2 1 (1+ ) 2 j +1 o k 2 X j =1 k 2 j 1 k 2 j +1 2 j 1 (1+ )(2 j ) (2 +1) n 2 1 (1+ ) 2 j o ; if k iseven, k k 1 2 X j =1 k 2 j k 2 j 2 j (1+ )(2 j +1) (2 +1) n 2 1 (1+ ) 2 j +1 o + k +1 2 X j =1 k 2 j 1 k 2 j +1 2 j 1 (1+ )(2 j ) (2 +1) n 2 1 (1+ ) 2 j o ; if k isodd. (5.2.4) 48

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where =E( X )istheexpectationof X .Itfollowsfrom(5.2.3)and(5.2.4)thatthe expectation,variance,skewnessandkurtosisof X aregivenby Exp( X )= (1+4 +2 2 ) (1+ )(1+2 ) Var( X )= 2 (1+8 +16 2 +12 3 +4 4 ) (1+ ) 2 (1+2 ) 2 Ske( X )= 2(1+12 +42 2 +70 3 +66 4 +36 5 +8 6 ) (1+8 +16 2 +12 3 +4 4 ) 3 = 2 and Kur( X )= 3(176 10 +1408 9 +4944 8 +10000 7 +12824 6 +10728 5 ) (1+ ) 2 (1+8 +16 2 +12 3 +4 4 ) 2 + (5800 4 +1992 3 +427 2 +54 +3) (1+ ) 2 (1+8 +16 2 +12 3 +4 4 ) 2 Agraphicalrepresentationofthesestatisticalexpressio nsaregiveningure5.2asa functionoftheparameter Notethatfor =0 ; =1wehaveExp( X )=1 ; Var( X )=1 ; Skewness( X )=2 ; andKurtosis( X )=9yieldthestandardexponentialdistribution.Alsoitis clearthat bothexpectationandvariancearerstincreasingandthend ecreasingfunctionsof Theexpectationincreasesfrom to1 : 17157 as convergesfrom0to 1 p 2 andthen decreasesto as goesto 1 .Ontheotherhandthevarianceincreasesfrom 2 to 1 : 202676857 2 as increasesfrom0to0 : 3512071921andthendecreasesto 2 as goesto 1 49

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0246810 1.001.051.101.15 lE (X) 0246810 1.001.101.20 lVariance(X) 0246810 1.801.902.00 lSkewness(X) 0246810 1015202530 lKurtosis(X) Behavior of Expectation,Variance,Skewness and Kurtosis Figure5.2:Behaviorofexpectation,variance,skewnessan dkurtosisofTSLdistribution50

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5.3MGFandCumulantsThemomentgeneratingfunctionofarandomvariable X isdenedby M ( t )= E (exp( tX )).When X hasthepdf(5.1.2)directintegrationyieldsthat M ( t )= (1+ ) (1+2 ) (1+2 t ) (1 t )(1+ t ) for t< 1 = .Thus,thecharacteristicfunctiondenedby ( t )= E (exp( itX )andthe cumulantgeneratingfunctiondenedby K ( t )=log( M ( t ))taketheforms ( t )= (1+ ) (1+2 ) (1+2 it ) (1 it )(1+ it ) and K ( t )=log 1+ 1+2 +log (1+2 t ) (1 t )(1+ t ) ; respectively,where i = p 1isthecomplexnumber.Byexpandingthecumulant generatingfunctionas K ( t )= 1 X k =1 a k ( t ) k k ; oneobtainsthecumulants a k givenby a k =( k 1)! k 1+ 1 (1+ ) k 1 (1+2 ) k 51

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5.4PercentilesAsmentioninchapter3wearealwaysinterestedtocomputeth epercentiles.The 100 p thpercentile x p isdenedby F ( x p )= p ,where F isgivenby(5.1.1).Then x p is thesolutionofthetranscendentalequation 1+ exp (1+ ) x p 2(1+ )exp x p (2 +1) = p Substituting y p = 2(1+ )exp( x p ) (1+2 )(1 p ) ,thisequationcanbereducedto 1 y p + (2 +1) (1 p ) [2(1+ )] 1+ y 1+ p =0 ; whichtakestheformof(3.1.14).Thus,using(3.1.15), y p isgivenby y p =1+ 1 X j =1 (1+ ) j j 1 1 j [(1+2 )(1 p )] j (2+2 ) (1+ ) j andhencethesolutionfor x p isgivenby x p = log ( (1 p )(1+2 ) 2(1+ ) 1+ 1 X j =1 1+ ) j j 1 1 j [(1+2 )(1 p )] j (2+2 ) (1+ ) j !) (5.4.5) 5.5MeanDeviationAsmentioninchapter3,ifweareinterestedtondtheamount ofscatterina populationisevidentlymeasuredtosomeextentbythetotal ityofdeviationsfrom themean.Thisisknownasthemeandeviationaboutthemeanan ditisdenedby 1 ( X )= Z 1 0 j x j f ( x ) dx 52

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where = E ( X ).Thismeasurescanbecalculatedusingtherelationshipst hat 1 ( X )= Z 0 ( x ) f ( x ) dx + Z 1 ( x ) f ( x ) dx: Thus,foraTSLrandomvariable X wehave 1 ( X )= 4 (1+ ) (1+2 ) exp (1+4 +2 2 ) (1+ )(1+2 ) 2 (1+ )(1+2 ) exp (1+4 +2 2 ) (1+2 ) : (5.5.6) 5.6EntropyAsmentioninchapter3,westudytheentropytomeasuretheva riationoftheuncertainty.Renyientropyisdenedby J R ( r )= 1 1 r log Z f r ( x ) dx ; (5.6.7) where r> 0and r 6 =1(Renyi,)[19].Forthepdf(5.1.2),notethat J R ( r )= 1 (1 r ) log (1+ ) (1+2 ) r + log Z 1 0 2exp( x ) exp( (1+ ) x ) r dx : Furthercalculationyieldstheentropyintermsofincomple tebetafunctionas J R ( r )= r 1 r log 1+ (1+2 ) + 1 1 r log 2 r (1+ 1 ) B 1 = 2 r ;r +1 : TheShanonentropyofaadistributionFisdenedby H ( F )= X i p i log p i where p 0i s aretheprobabilitiescomputedfromthedistributionF.How ever,theextensionofthediscretecasetothecontinuouscasewithdist ribution F anddensity f 53

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iscalleddierentialentropyandisgivenby H ( F )= E ( log f ( x ))= Z f ( x )log f ( x ) dx HoweverthisextensionhasafewdrawbacksaspointedoutbyR ao etal. [23],like, itmayassumeanyvalueontheextendedrealline,itisdened forthedistributions withdensitiesonly.Rao etal. [23]introducedthe cumulativeresidualentropy (CRE)denedby E ( X )= Z Pr( j X j >x )logPr( j X j >x ) dx; whichismoregeneralthanShannon'sentropyinthattheden itionisvalidinthe continuousanddiscretedomains.Forthepdf(5.1.2),wecan write Pr( j X j >x )= 2(1+ )exp( x ) exp( (1+ ) x ) (1+2 ) : Hence,E ( X )= 1 (1+2 ) Z 1 0 exp( (1+ ) x )log 2(1+ )exp( x ) exp( (1+ ) x ) (1+2 ) dx 2(1+ ) (1+2 ) Z 1 0 exp( x )log 2(1+ )exp( x ) exp( (1+ ) x ) (1+2 ) dx: UsingTaylorexpansionandonintegratingwehavetheCREgiv enby E ( X )= (1+6 +6 2 +2 3 ) (1+ ) 2 (1+2 ) + (1+4 +2 2 )log(1+2 ) (1+ )(1+2 ) (1+2 ) 1 X k =1 2 k (1+ ) k k ( k + +1) + 2 (1+ ) (1+2 ) 1 X k =1 2 k (1+ ) k k ( k +1) (1+4 +2 2 )log(2+2 ) (1+ )(1+2 ) (5.6.8) 54

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Notethat E ( X )in(5.6.8)ispositiveandeachseriesisconvergentandisb ounded aboveby1.Thus,wehave E ( X ) (1+6 +6 2 +2 3 ) (1+ ) 2 (1+2 ) + (1+4 +2 2 )log(1+2 ) (1+ )(1+2 ) + 2 (1+ ) (1+2 ) : 5.7EstimationGivenarandomsample X 1 ;:::;X n from(5.1.2),weareinterestedinestimatingthe inherentparametersusingthemethodofmoments.Byequatin gthetheoreticalexpressionsfor E ( X )and E ( X 2 )withthecorrespondingsampleestimates,oneobtains theequations: m 1 = (1+4 +2 2 ) (1+ )(1+2 ) (5.7.9) and m 2 =2 2 (1+6 +6 2 +2 3 ) (1+ ) 2 (1+2 ) (5.7.10) where m 1 = 1 n n X i =1 x i and m 2 = 1 n n X i =1 x 2i : Onsolvingthissystemofequationswehave(8 m 21 4 m 2 ) 4 +(28 m 21 16 m 2 ) 3 +(36 m 21 20 m 2 ) 2 +(16 m 21 8 m 2 ) +(2 m 21 m 2 )=0 whichcontainstheparameter only.Onecansolvethisequationfor .Usingeither (5.7.9)or(5.7.10)wecangetthecorrespondingvalueof 55

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Themethodofmaximumlikelihoodtoestimatetheparameters isdescribedbelow: Theunderlyinglikelihoodfunctionforacompletesampleof size n isgivenby L ( ; ; x 1 ;x 2 ;:::x n )= (1+ ) n n (1+2 ) n n Y i =1 2exp( x i ) exp( (1+ ) x i ) (5.7.11) Thecorrespondinglog-likelihoodfunctionisgivenby:ln L ( ; )= n ln(1+ ) n ln n ln(1+2 ) 1 n X i =1 x i + n X i =1 ln[2 exp( x i )] : Takingthederivativeoftheabovefunctionwithrespectto and andequatingeach equationstozeroweobtain n 1+ 2 n 1+2 + 1 n X i =1 x i exp( x i ) [2 exp( x i )] =0(5.7.12) and n + 1 2 n X i =1 x i 2 n X i =1 x i exp( x i ) [2 exp( x i )] =0 : (5.7.13) Theseequationscannotbesolvedanalyticallybutstatisti calsoftwarecanbeusedto ndthemaximumlikelihoodestimatorsof and 56

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5.8ReliabilityandHazardRateFunctionsThesubjectpdfcanbeausefulcharacterizationoffailuret imeofagivensystem becauseoftheanalyticalstructure.Thus,Thereliability function R ( t ),whichisthe probabilityofanitemnotfailingpriortosometime t ,isdenedby R ( t )=1 F ( t ). ThereliabilityfunctionforTSL( ; )probabilitydistributionisgivenby R ( t )= 2(1+ )exp t exp (1+ ) t (2 +1) : (5.8.14) Thehazardratefunction,alsoknownasinstantaneousfailu reratefunctionis denedby h ( t )=lim t 0 Pr ( tt ) t = f ( t ) R ( t ) : Itisimmediatefrom(5.1.2)and(5.1.1)thatthehazardrate functionforTSLdistributionisgiven h ( t )= (1+ ) n 2 exp t o n 2+2 exp t o : Itisclearthat h isanincreasingfunction,itincreasesfrom 1 (1+ ) (1+2 ) to 1 astvaries from0to 1 Thisisanimportantfeatureofthispdfwhichmakesitquited ierentfromthe exponentialandWeibullpdf'swherethehazardrateforanex ponentialdistributionis constantwhereasthehazardrateforWeibulldistributioni seitherstrictlyincreasing orstrictlydecreasing.Inthepresentstudywewillpresent acomparisonsofTSL distributionwithsomeotherlifetimedistributionwhosep robabilitydensityfunction aresimilartothatofTSLdistribution. 57

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ThecumulativehazardratefunctionforTSLisgivenby H ( t )= Z t 0 h ( u ) du = log( R ( t )) = t +log 1+2 2+2 exp( t ) : Alsowehave A ( t )= 1 t Z t 0 h ( ) d; asthefailure(hazard)rateaverage.Thus,foraTSLrandomv ariablethefailurerate averageisgivenby A ( t )= 1 + 1 t log 1+2 2+2 exp( t ) : Agraphicalrepresentationofthereliability R ( t )andthehazardrate h ( t )for =1 andvariousvaluesoftheparameter isgiveninFigure5.3. NotethatTSLhasincreasingfailurerate(IFR)hencetherel iabilityfunctionisdecreasing.Alsonotefromtheguresthatsignicancediere ncesoccurattheearly time.Infactthehazardrateisconstantfor =0whichmeansexponentialpdfis aparticularcaseofTSLpdf.Moreover,foragivenvalueofth eparameter ,the reliabilityincreasesuntilitattains =1andthenitdecreases. 58

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0246810 0.00.20.40.60.81.0 tR(t) 0246810 0.00.20.40.60.81.0 0246810 0.00.20.40.60.81.0 0246810 0.00.20.40.60.81.0 0246810 0.00.20.40.60.81.0 l= 0 l= 1 l= 2 l= 10 l= 50 Relaibility for TSL Distribution 0246810 0.60.70.80.91.0 th(t) 0246810 0.60.70.80.91.0 0246810 0.60.70.80.91.0 0246810 0.60.70.80.91.0 0246810 0.60.70.80.91.0 0246810 0.60.70.80.91.0 l= 0 l= 0.5 l= 1 l= 2 l= 5 l= 10 Hazard rate for TSL Distribution Figure5.3:ReliabilityandhazardrateofTSLdistribution for =159

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5.9MeanResidualLifeTimeandtheMeanTimeBetweenFailureThemeanresiduallife(MRL)atagiventime t measurestheexpectedremaininglife timeofanindividualofage t .Itisdenotedby m ( t )andisdenedas m ( t )= E ( T t j T t ) = R 1 t R ( u ) du R ( t ) Thecumulativehazardratefunctionisgivenby H ( t )= log( R ( t ))andwecan expressthemeanresiduallifetimeintermsof H by m ( t )= Z 1 0 exp( H ( t ) H ( t + x )) dx (5.9.15) Now,ifweconsidertheconverseproblem,thatofexpressing thefailurerateinterms ofthemeanresiduallifeanditsderivativeswehave m 0 ( t )= h ( t ) m ( t ) 1 : (5.9.16) Hence,theMRLforaTSLrandomvariableisgivenby m ( t )= (1+ ) ( 2(1+ ) 2 exp( t ) 2(1+ ) exp( t ) ) (5.9.17) Now,wediscussthemeantimebetweenfailure(MTBF)forthet runcatedskewLaplacepdf.Thetimedierencebetweentheexpectednextfa iluretimeandcurrent failuretimeiscalledthe MeanTimeBetweenFailure (MTBF).Manyscientistsand engineersconsiderthereciprocaloftheintensityfunctio n(alsocalledthehazardrate function)atcurrentfailuretimeastheMTBF.Thatis, MTBF = 1 ( t ) (5.9.18) where ( t )istheintensityfunction.BasedonthisdenitiontheMTBF forTSL 60

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distributionwillbegivenby MTBF = (1+ ) n 2(1+ ) exp t o n 2 exp t o : (5.9.19) ButtheMeantimebetweenfailureisindeedtheexpectedinte rvallengthfromthe currentfailuretime,say T n = t n ,tothenextfailuretime, T n +1 = t n +1 .Wewilluse MTBF n todenotetheMTBFatcurrentstate T n = t n .Hence,itfollowsthat MTBF n = Z 1 t n tf n +1 ( t j t 1 ;t 2 ;:::;t n ) dt t n (5.9.20) where R 1 t n tf n +1 ( t j t 1 ;t 2 ;:::;t n ) dt istheexpected( n +1)thfailureunderthecondition T n = t n ForTSLdistribution,wehave f n +1 ( t j t 1 ;t 2 ;:::;t n )= 1 2(1+ )exp( t ) (1+ )exp( (1+ ) t ) 2(1+ )exp( t n ) exp( (1+ ) t n ) (5.9.21) Hence,the MTBF n isgivenby MTBF n = [2(1+ )exp( t n ) 1 (1+ ) exp( (1+ ) t n )] [2(1+ )exp( t n ) exp( (1+ ) t n )] = (1+ ) n 2(1+ ) 2 exp( t n ) o n 2(1+ ) exp( t n ) o : (5.9.22) Itisclearthatforthespecialcase =0wehave MTBF n = Thus,inthecase =0,wehave MTBF n = 1 n (5.9.23) Notethat,inthisspecialcasetheprocesshasaconstantint ensityfunction.Forother valuesof wealwayshave MTBF n < 1 n 61

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5.10ConclusionInthischapterwehavestudiedallanalyticalaspectsofTSL pdf.Wehavederived theanalyticalformofmoments,momentgeneratingfunction ,cumulantgenerating function,cumulants,percentilesandentropyifarandomva riablefollowstheTSLpdf. Inadditionwehavedevelopedthecorrespondingestimation sformsofthesubject parameters.Also,wehavedevelopedthereliabilitymodela nditscorresponding hazardratefunctionwhenthefailuretimesarecharacteriz edbyTSLpdf.Inaddition, wehavedevelopedtheanalyticalformsofmeanresiduallife timesandmeantime betweenfailuresanditsrelationshiptotheintensityfunc tionforsystemsthatexhibit thecharacteristicofsubjectpdf. 62

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Chapter6 ComparisonofTSLProbabilityDistributionwithOtherDist ributions 6.1IntroductionInthepresentstudywewilldiscussacomparisonofthetrunc atedskewLaplacepdf withsomeotherpopularpdf,namely,thegammapdfandthehyp oexponentialpdf whosegraphicalrepresentationandcharacterizationares imilartothatofTSLpdf. Insection6.2wewillconsiderthecomparisonswithtwopara metergammamodel andinsection6.3wewillbecomparingTSLpdfwiththesocall edhypoexponential pdf.Wewillmakecomparisonsintermsofthereliabilitybeh avior.Infact,wewill simulatedatafromTSLpdfandcheckwhetherithasthesamere liabilityifthedata wasassumedtobegammapdfandhypoexponentialpdf.Alsowew illconsideradata consistingofthefailuretimeofpressurevesselsdatawhic hwasstudiedbyKeating etal. [13]usinggammapdf. 6.2TSLVs.TwoParameterGammaProbabilityDistributionThegammaprobabilitydistributionplaysacrucialroleinm athematicalstatistics andmanyappliedareas.Arandomvariable X issaidtohavegammaprobability distributionwithtwoparameters and ,denotedby G ( ; ),if X hastheprobability densityfunctiongivenby f ( x ; ; )= 1 ( ) t 1 exp( t ) ;;;t> 0 : 63

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where( )denotesthegammafunctionevaluatedat .Theparameters and are theshapeandscaleparameters,respectively.Thereliabil ityandhazardfunctionsare notavailableinclosedformunless happenstobeaninteger;however,theymaybe expressedintermsofthestandardincompletegammafunctio n( a;z )denedby ( a;z )= Z z 0 y a 1 exp( y ) dy;a> 0 : Intermsof( a;z ),thereliabilityfunctionforthe G ( ; )distributionisgivenby R ( t ; ; )= ( ) ( ;t= ) ( ) ; and,if isaninteger,itisgivenby R ( t ; ; )= 1 X k =0 ( t= ) k exp( t= ) k : Thehazardrateisgivenby h ( t ; ; )= t 1 exp( t= ) [( ) ( ;t= )] ; forany > 0and,if isanintegeritbecomes h ( t;; )= t 1 ( ) P 1 k =0 ( t= ) k =k : Theshapeparameter isofspecialinterestsincewhether 1isnegative,zero orpositive,correspondstoadecreasingfailurerate(DFR) ,constant,orincreasing failurerate(IFR),respectively.Itisclearthatthegammamodelhasmorerexibilitythanthat ofTSLmodelasthe formeronecanbeusedevenifthedatahasDFR.Infact,theexp onentialmodel isaparticularcaseofbothmodels,thatis, TSL (0 ; 1)and Gamma (1 ; 1)arethe exponentialmodels.Butifinthegammamodel > 1,ithasIFRwhichappearsto bethesameasthatofTSLmodelbutacarefulstudyshowsthatt hereisasignicant dierenceinthesetwomodelseveninthiscase. 64

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Figure6.1givesagraphicalcomparisonsofthereliability functionsofTSLand Gammapdf.Itisclearlyseenthat Gamma (1 ; 1)and TSL (0 ; 1)areidenticalyielding theexponential EXP (1)model. 0246810 0.00.20.40.60.81.0 tR(t) 0246810 0.00.20.40.60.81.0 0246810 0.00.20.40.60.81.0 0246810 0.00.20.40.60.81.0 TSL(0 1)TSL(1 1)GAMMA(1 1)GAMMA(1.32 0.93) Reliability of TSL and Gamma distributions Figure6.1:ReliabilityofTSLandGammadistributions65

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Table6.1givesacomparisonsoftwoparametergammamodelwi threspecttoTSL model.Wesimulate100datafordierentvaluesoftheTSLpar ameters and as indicatedinthetable.WeusedtheNewton-Raphsonalgorith mtogetitsestimates. Againwegettheestimatesoftheparameters and ofgammamodelassumingthat thedatasatisfythegammamodel. b b b b n1 n2 1 0 1 : 045522 0 : 06166375 1 : 013032 1 : 085437 4 4 1 1 1 : 056979 1 : 341615 1 : 317944 0 : 9267452 3 7 1 2 1 : 067631 1 : 478364 1 : 379657 0 : 8905008 5 7 1 5 1 : 012352 4 : 362386 1 : 231728 0 : 8906503 6 6 1 10 1 : 011322 9 : 99997 1 : 189242 0 : 8872343 6 6 2 0 2 : 081052 0 : 000026 0 : 9989892 2 : 083215 5 5 2 1 2 : 055891 0 : 91154 1 : 380973 1 : 74020 6 8 2 2 2 : 098627 2 : 024318 1 : 313875 1 : 809048 5 6 2 5 1 : 965912 5 : 942018 1 : 20621 1 : 738104 6 6 2 10 2 : 10125 10 : 92677 1 : 250668 1 : 747456 4 4 5 0 4 : 619639 0 : 6206207 1 : 126453 4 : 801781 7 7 5 1 5 : 45453 1 : 144246 1 : 378976 4 : 597368 6 8 5 2 5 : 128795 1 : 554738 1 : 358207 4 : 335356 5 6 5 5 4 : 817129 4 : 976809 1 : 223907 4 : 235068 5 5 5 10 5 : 488563 9 : 635667 1 : 238111 4 : 631146 3 3 Table6.1:ComparisonbetweenTSLandgammamodelswhenboth parametersareunknownInTable6.1wehaveusedtheKolmogrov-Smirnovnon-paramet rictesttocheck whetherthedatageneratedfromTSL( ; )alsosatisedtheGamma( ; ).The tableshowsthattheTSLpdfcloselyresemblesatwoparamete rgammapdf.Inthe table n 1 and n 2 ,respectively,denotethecorrespondingnumberofitemfai ledbefore T T and T G .Where T T and T G aredenedby P ( T T T ) 0 : 95 and P ( T T G ) 0 : 95 66

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respectively.Notethat T T and T G respectivelydenotethefailuretimeassumingTSL andGammapdf.Table6.1showsthatthesignicancedierenceoccurswhenp arameter =1and theyareidenticalfor =0.Alsoif > 1thenbiggerthevalueof closerthe relationwithgammapdfsubjecttotheconditionthatthepar ameter remainsthe same. Ifthetwomodelshappentobeidenticalweshouldbeableton dtheparameters ofonedistributionknowingtheparametersoftheother.Aus ualtechniqueisby equatingthersttwomoments.Ifthisisthecasewemusthave thefollowingsystem ofequations 8><>: (2 2 1) 2 2 = 2 2 (2 3 1) 2 3 2 = 2 2 + 2 where =1+ : Onsolvingthesystemofequationsweget = (2 2 1) 2 (4 4 4 3 +4 2 +1) ; and = (4 4 4 3 +4 2 +1) (2 2 )(2 2 1) : Thisshowsthatonecangettheparameters and ofaGammadistributionifwe knowtheparameters and ofaTSLdistribution. Ifweknoworhavedatatoestimatetheparameter thentheMLEof isgiven by b = (2 +1)( +1) n (2 2 +4 +1) n X i =1 X i AgainwecomparetheTSLpdfwithgammapdfassumingthatthep arameter isknown.Wefollowthesameprocedureasinthepreviouscase andtheTable6.2 67

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alsopresentsthesamequantitiesasTable6.1does. b b b n1 n2 1 1 1 : 011927 1 : 200189 0 : 9836633 4 5 1 2 1 : 009196 1 : 102708 1 : 037224 4 4 1 5 0 : 9884295 1 : 111627 0 : 9565353 2 2 1 10 1 : 010180 1 : 166273 0 : 9036562 3 3 1 50 0 : 923165 1 : 015271 0 : 9181059 5 5 2 1 1 : 812826 1 : 168546 1 : 809911 2 3 2 2 2 : 013709 1 : 190309 1 : 917320 5 6 2 5 2 : 031695 1 : 186196 1 : 842538 5 5 2 10 2 : 008019 1 : 219654 1 : 717656 2 2 2 50 1 : 835290 1 : 217294 1 : 522315 4 4 5 1 5 : 45453 1 : 378976 4 : 597368 6 8 5 2 5 : 128795 1 : 358207 4 : 335356 5 6 5 5 4 : 817129 1 : 223907 4 : 235068 5 5 5 10 5 : 488563 1 : 238111 4 : 631146 3 3 5 50 4 : 716769 1 : 043489 4 : 565573 5 5 Table6.2:ComparisonbetweenTSLandgammamodelswhenonep arameterisknownTable6.2alsosupportstheconclusionwehavedrawnfromthe previoustable. Thatmeansthereisasignicancedierencebetweenthesetw omodelswhentheparameter =1.Butforalargevalueof wedonotseemuchdierencesintermsof reliabilitysubjecttotheconditionthevalueoftheparame ter remainsthesame. Finally,wewouldliketopresentarealworldproblemwheret heTSLmodelgives abettertthanthecompetinggammamodel.Thefollowingdataisthefailuretimes(inhours)ofpressur evesselsconstructedof ber/epoxycompositematerialswrappedaroundmetallines subjectedtoacertain constantpressure.ThisdatawasstudiedbyKeating etal. [12]. 274,1.7,871,1311,236 458,54.9,1787,0.75,77628.5,20.8,363,1661,828 290,175,970,1278,126 68

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Pal,N. etal. [20]mentionthatthe Gamma (1 : 45 ; 300)modeltsforthesubject data.WehaverunaKolmogoroveSmirnovnonparametricstati sticaltestandobservedthefollowingresults:K-Sstatistics D Gamma =0 : 2502and D TSL =0 : 200forGamma(1.45,300)andTSL( 5939.8,575.5)distributionrespectively. Figure6.2exhibitsthep-pplotofthepressurevesseldataa ssumingTSLand Gammapdf.ItisevidentthatTSLtsbetterthantheGammamod el.Hencewe recommendthatTSLisabettermodelforthepressurevesseld ata. 0.00.20.40.60.81.0 0.00.20.40.60.81.0 P-P plot of Pressure Vessels DataObservedExpected 0.00.20.40.60.81.0 0.00.20.40.60.81.0 TSL GAAMA Figure6.2:P-PPlotsofVesseldatausingTSLandGammadistr ibution69

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Table6.3belowgivesthereliabilityestimatesusingTSLpd fandGammapdfand Figure6.3exhibitsthereliabilitygraphs.Thereisasigni cancedierencesonthe estimates. t b RTSL( t ) b RGAMMA( t ) t b RTSL( t ) b RGAMMA( t ) 0 : 75 0 : 999 0 : 999 363 0 : 532 0 : 471 1 : 70 0 : 997 0 : 999 458 0 : 451 0 : 365 20 : 80 0 : 965 0 : 984 776 0 : 260 0 : 150 28 : 50 0 : 952 0 : 976 828 0 : 237 0 : 129 54 : 90 0 : 909 0 : 940 871 0 : 220 0 : 113 126 : 0 0 : 803 0 : 826 970 0 : 185 0 : 085 175 : 0 0 : 738 0 : 745 1278 0 : 108 0 : 034 236 : 0 0 : 664 0 : 647 1311 0 : 102 0 : 030 274 : 0 0 : 621 0 : 590 1661 0 : 056 0 : 010 290 : 0 0 : 604 0 : 567 1787 0 : 045 0 : 007 Table6.3:TheReliabilityestimatesofPressureVesselsDa ta 050010001500 0.00.20.40.60.81.0 Reliability of the Pressure Vessel DatatR(t) 050010001500 0.00.20.40.60.81.0 TSLGAMMA Figure6.3:ReliabilityofVesseldatausingTSLandGammadi stribution70

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6.3TSLVs.HypoexponentialProbabilityDistributionObservingtheprobabilitystructureofthetruncatedskewL aplacepdfitisourinteresttolookforanexistingprobabilitydistributionwhichc anbewrittenasadierence oftwoexponentialfunction.WewillcompareTSLpdfwiththe hypoexponentialpdf. Manynaturalphenomenoncanbedividedintosequentialphas es.Ifthetimetheprocessspendsineachphaseisindependentandexponentiallyd istributed,thenitcanbe shownthattheoveralltimeishypoexponentiallydistribut ed.Ithasbeenempirically observedthattheservicetimesforinput-outputoperation sinacomputersystem oftenpossessthisdistributionseeK.S.Trivedi[25].Itwi llhave n parametersone foreachofitsdistinctphases.Herweareinterestedinatwo -stagehypoexponential process.Thatis,ifXbearandomvariablewithparameters 1 and 2 ( 1 6 = 2 ),then itspdfisgivenby f ( x )= 1 2 2 1 f exp( 1 x ) exp( 2 x ) g x> 0 : Wewillusethenotation Hypo ( 1 ; 2 )todenoteahypoexponentialrandomvariable withparameters 1 and 2 ,respectively.Figure6.4givesagraphicaldisplayofthe pdfofthehypoexponentialdistributionfor 1 =1anddierentvaluesof 2 > 1 .In fact,becauseofthesymmetryitdoesn'tmatterwhichparame terneedtobebigger andwhichonetobesmaller.Figure5.4hastheparameters 1 =1and 2 =1 : 5 ; 2 ; 3 ; 5 ; 10 ; 50.Fromthegureit isclearlyseenthatasthevalueoftheparameter 2 increasesthepdflookslikeTSL pdf.Thecorrespondingcdfisgivenby F ( x )=1 2 2 1 exp( 1 x )+ 1 2 1 exp( 2 x ) x 0 : TheReliabilityfunction R ( t )ofa Hypo ( 1 ; 2 )randomvariableisgivenby R ( t )= 2 2 1 exp( 1 t ) 1 2 1 exp( 2 t ) : (6.3.1) 71

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0246810 0.00.20.40.60.8 xf(x) 0246810 0.00.20.40.60.8 0246810 0.00.20.40.60.8 0246810 0.00.20.40.60.8 0246810 0.00.20.40.60.8 0246810 0.00.20.40.60.8 l2= 1.5 l2= 2 l2= 3 l2= 5 l2= 10 l2= 50 PDF of Hypoexponential Distribution Figure6.4:PDFofHypoexponentialDistributionfor 1=1anddierentvaluesof 2 Thehazardratefunction h ( t )ofa Hypo ( 1 ; 2 )randomvariableisgivenby h ( t )= 1 2 [exp( 1 t ) exp( 2 t )] 2 exp( 1 t ) 1 exp( 2 t ) : (6.3.2) Itisclearthat h ( t )isincreasingfunctionoftheparameter 2 .Itincreasesfrom0 tomin f 1 ; 2 g .Figure6.5exhibitsthereliabilityfunctionandhazardha zardrate functionofahypoexponentialrandomvariablewithparamet ers 1 =1anddierent valuesof 2 > 1 72

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0246810 0.00.20.40.60.81.0 tR(t) 0246810 0.00.20.40.60.81.0 0246810 0.00.20.40.60.81.0 0246810 0.00.20.40.60.81.0 0246810 0.00.20.40.60.81.0 l2= 1.5 l2= 2 l2= 5 l2= 10 l2= 50 Reliability of Hypoexponential Distribution 0246810 0.00.20.40.60.81.0 th(t) 0246810 0.00.20.40.60.81.0 0246810 0.00.20.40.60.81.0 0246810 0.00.20.40.60.81.0 0246810 0.00.20.40.60.81.0 0246810 0.00.20.40.60.81.0 l 2 = 1.5 l 2 = 2 l 2 = 3 l 2 = 5 l 2 = 10 l 2 = 50 Hazard rate for hypoexponential Distribution Figure6.5:ReliabilityandhazardratefunctionofHypoexp onentialdistribution73

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Also,notethatthemeanresiduallife(MRL)timeattime t for Hypo ( 1 ; 2 )is givenby m Hypo ( t )= 1 1 2 22 exp( 1 t ) 21 exp( 2 t ) [ 2 exp( 1 t ) 1 exp( 2 t )] : WenowproceedtomakeacomparisonbetweenTSLandhypoexpon entialpdfin termsofthereliabilityandthemeanresiduallifetimes.We willgeneratefromthe hypoexponentialdistributionsarandomsamplesofsize50, 100and500fordierent valuesoftheparameters 1 2 andthenproceedtotthedatatoTSLmodel. n 1 2 c 1 c 2 b b MTSL MHYPO 50 1 2 0 : 934 2 : 325 2 : 745 1 : 349 1 : 362 1 : 129 50 1 5 0 : 975 5 : 133 2 : 779 1 : 097 1 : 108 1 : 029 50 1 10 0 : 979 12 : 223 1 : 042 6 : 968 1 : 042 1 : 021 50 1 20 0 : 940 26 : 742 1 : 069 15 : 349 1 : 069 1 : 063 100 1 2 0 : 876 2 : 565 1 : 376 2 : 403 1 : 391 1 : 184 100 1 5 0 : 903 6 : 835 1 : 178 6 : 216 1 : 179 1 : 108 100 1 10 0 : 950 9 : 838 1 : 098 8 : 439 1 : 099 1 : 052 100 1 20 1 : 029 26 : 322 0 : 892 0 : 242 0 : 982 0 : 971 500 1 2 0 : 915 2 : 576 1 : 339 3 : 076 1 : 348 1 : 132 500 1 5 0 : 961 6 : 489 1 : 088 3 : 453 1 : 093 1 : 042 500 1 10 0 : 881 10 : 224 1 : 174 8 : 355 1 : 173 1 : 135 500 1 20 1 : 016 27 : 411 0 : 988 14 : 044 0 : 988 0 : 983 Table6.4:ComparisonbetweenTSLandHypoexponentialMode lsTocreateTable6.4wegeneratearandomsampleofsizes50,10 0and500from hypoexponentialpdfwithparameters 1 and 2 =2 ; 5 ; 10&20foreachsamplesize. WehaveusedtheNewton-Raphsonalgorithmtoestimatethema ximumlikelihood estimatorsof 1 and 2 .WeexpectthatthedatawillttheTSLmodel,soassuming thedatasatisesTSLmodelweestimatetheparameters and Inadditionwecomputedthemeanresiduallifetimesforboth themodelsat T = T n= 2 Fromthetablewecanobservethatifthesamplesizeislargea ndthedierencebetweenthetwoparameters 1 and 2 islargebothTSLandhypoexponentialmodel willproducethesameresultwhereasforsmallsizeandsmall dierencebetween 1 74

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and 2 thereisasignicantdierences.Thegures(6.6-6.8)belo walsosupportthis argument. 012345 0.00.20.40.60.81.0 Reliability for TSL and Hypoexponential Distributions(a) tR(t) 012345 0.00.20.40.60.81.0 HypoexponentialTSL 012345 0.00.20.40.60.81.0 (b) tR(t) 012345 0.00.20.40.60.81.0 HypoexponentialTSL 01234 0.00.20.40.60.81.0 (c) tR(t) 01234 0.00.20.40.60.81.0 HypoexponentialTSL 01234 0.00.20.40.60.81.0 (d) tR(t) 01234 0.00.20.40.60.81.0 HypoexponentialTSL Figure6.6:ReliabilityofTSLandHypoexponentialdistrib utionsfor n =50 ;1=1 ; and( a ) 2=2 ; ( b ) 2=5 ; ( c ) 2=10 ; ( d ) 2=2075

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012345 0.00.20.40.60.81.0 Reliability for TSL and Hypoexponential Distribution(a) tR(t) 012345 0.00.20.40.60.81.0 HypoexponentialTSL 012345 0.00.20.40.60.81.0 (b) tR(t) 012345 0.00.20.40.60.81.0 HypoexponentialTSL 012345 0.00.20.40.60.81.0 (c) tR(t) 012345 0.00.20.40.60.81.0 HypoexponentialTSL 01234 0.00.20.40.60.81.0 (c) tR(t) 01234 0.00.20.40.60.81.0 HypoexponentialTSL Figure6.7:ReliabilityofTSLandHypoexponentialdistrib utionsfor n =100 ;1=1 ; and( a ) 2=2 ; ( b ) 2=5 ; ( c ) 2=10 ; ( d ) 2=2076

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01234567 0.00.20.40.60.81.0 Reliability for TSL and Hypoexponential Distribution(a) tR(t) 01234567 0.00.20.40.60.81.0 HypoexponentialTSL 01234567 0.00.20.40.60.81.0 (b) tR(t) 01234567 0.00.20.40.60.81.0 HypoexponentialTSL 02468 0.00.20.40.60.81.0 (c) tR(t) 02468 0.00.20.40.60.81.0 HypoexponentialTSL 0123456 0.00.20.40.60.81.0 (d) tR(t) 0123456 0.00.20.40.60.81.0 HypoexponentialTSL Figure6.8:ReliabilityofTSLandHypoexponentialdistrib utionsfor n =500 ;1=1 ; and( a ) 2=2 ; ( b ) 2=5 ; ( c ) 2=10 ; ( d ) 2=2077

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6.4ConclusionInthischapterwehavecomparedtheTSLmodelwithtwocommon lyusedexisting modelsnamely,theGammamodelandhypoexponentialmodelsi ntermsoftheir reliabilitybehavior.Wehaveseenthatboth TSL (0 ; 1)and Gamma (1 ; 1)areidentical asbothyieldtheexponentialmodelbutacarefulstudyshows therearesomesituations wheretheTSLmodelgivesabettertthangammamodel.Indeed wehaveillustrate thiswitharealinformationofPressureVesselsfailuredat a. Alsowemakeacomparisonwiththehypoexponentialpdfandha veconcludedthatif thesamplesizeislargeandthedierencebetweenthetwopar ameters 1 and 2 is alsolargebothTSLandhypoexponentialmodelwillproducet hesameresultwhereas forsmallsizeandsmalldierencebetween 1 and 2 thereisasignicantdierences. InfacttheshapeofthehazardratefunctionoftheTSLmodels eemslikeagraphof afunctionoftheform h ( t )=1 exp( t ).Workingbackward,wehavederivedthe correspondingpdfasshownbelow. SincetherelationbetweenthehazardratefunctionandtheC DFcanbeexpressed intermsofadierentialequationgivenby h ( t )= d dt log(1 F ( t )) ; andonsolvingthisequationwehave 1 F ( t )=exp Z t 0 h ( u ) du whichyieldsanewdistributionwhosepdfandCDFarerespect ively f ( t )=(1 exp( t ))exp 1 (1 exp( t )) t (6.4.3) and F ( t )=1 exp 1 (1 exp( t )) t (6.4.4) 78

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Figurebelowshowstheprobabilitydistributionfunctiono fthisdistributionfor dierentvaluesof 0246810 0.00.20.40.60.8 xf(x) 0246810 0.00.20.40.60.8 0246810 0.00.20.40.60.8 0246810 0.00.20.40.60.8 0246810 0.00.20.40.60.8 0246810 0.00.20.40.60.8 a= 0.1 a= 0.5 a= 1 a= 2 a= 5 a= 10 pdf of the distribution derived from hazard rate 79

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Chapter7 PreventiveMaintenanceandtheTSLProbabilityDistributi on 7.1IntroductionInmanysituations,failureofasystemorunitduringactual operationcanbevery costlyorinsomecasesquitedangerousifthesystemfails.T hus,itisbettertorepair orreplacebeforeitfails.Butontheotherhand,onedoesnot wanttomaketoo frequentreplacementofthesystemunlessitisabsolutelyn ecessary.Thuswetryto developareplacementpolicythatbalancesthecostoffailu resagainstthecostof plannedreplacementormaintenance.Supposethataunitwhichistooperateoveratime0totimet,[ 0,t]isreplaced uponfailure(withfailureprobabilitydistributionF).We assumethatthefailures areeasilydetectedandinstantlyreplaced.Acost c 1 thatincludesthecostresulting fromplannedreplacementandacost c 2 thatincludesallcostsresultingfromfailure isinvested.Thentheexpectedcostduringtheperiod[0,t]i s C ( t )= c 1 E ( N 1 ( t ))+ c 2 E ( N 2 ( t )) ; where E ( N 1 ( t ))and E ( N 2 ( t ))denotestheexpectednumberofplannedreplacement andexpectednumberoffailures.Wewouldliketoseekthepolicyminimizing C ( t )foranitetimespanorminimizing lim t !1 C ( t ) t foraninnitetimespan.SincetheTSLprobabilitydistribu tionhasan increasingfailurerateweexceptthismodeltobeusefulinm aintenancesystem. 80

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7.2AgeReplacementPolicyandTSLProbabilityDistributio n Firstweconsiderthesocalledthe\Agereplacementpolicy" .Inthispolicywe alwaysreplaceanitemexactlyatthetimeoffailureor t hoursafteritsinstallation, whicheveroccursrst.Agereplacementpolicyforaninnit etimespanseemstohave receivedthemostattentionintheliterature.Morese(1958 )showedhowtodetermine thereplacementintervalminimizingcostperunittime.Bar low etal. [5]provedthat ifthefailuredistribution,F,iscontinuousthenthereexi stsaminimum-costage replacementforanyinnitetimespan.Herewewouldliketodeterminetheoptimal t atwhichpreventivereplacementshould performed.Themodeldeterminesthe t thatminimizesthetotalexpectedcostof preventiveandfailuremaintenanceperunittime.Thetotal costpercycleconsists ofthecostofpreventivemaintenanceinadditiontothecost offailuremaintenance. Hence, EC ( t )= c 1 ( R ( t ))+ c 2 (1 R ( t ))(7.2.1) where, c 1 and c 2 denotethecostofpreventivemaintenanceandfailuremaint enance respectively. R ( t )istheprobabilitytheequipmentsurvivesuntilage t .Theexpected cyclelengthconsistsofthelengthofapreventivecycleplu stheexpectedlengthofa failurecycle.Thus,wehave Expectedcyclelength= t R ( t )+ M ( t )(1 R ( t ))(7.2.2) where, M ( t )(1 R ( t ))= Z t 1 tf ( t ) dt isthemeanofthetruncateddistributionattime t : Hence,the Expectedcostperunittime= c 1 R ( t )+ c 2 [1 R ( t )] t R ( t )+ M ( t )[1 R ( t )] : (7.2.3) 81

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Weassumethatasystemhasatimetofailuredistributionbei ngthetruncatedskew Laplacepdf.Wewouldliketocomputetheoptimaltime t ofpreventivereplacement. Hence,wehave R ( t )= 2(1+ )exp t exp (1+ ) t (2 +1) (7.2.4) and M ( t )= 1 1 R ( t ) Z t 0 tf ( t ) dt (7.2.5) Thus,wecanwrite Z t o tf ( t ) dt = 2 1 2 1 1 [1 exp( t = )] (2 1 1) 1 [1 exp( 1 t = )] + t (2 1 1) exp( 1 t = ) 2 1 t 2 1 1 exp( t = ) : (7.2.6) where 1 = +1. Onsubstitutingfrom(7.2.4),(7.2.5)and(7.2.6)in(7.2.3 )andsimplifyingtheexpressionswegettheexpectedcostperunittime(ECU)givenby ECU( t )= 1 f 2 1 ( c 2 c 1 )exp( t = ) ( c 2 c 1 )exp( 1 t = ) c 2 (2 1 1) g f 2 21 exp( t = ) 2 21 1 exp( 1 t = ) g Nowwewanttondthevalueof t whichminimizestheaboveexpressionsubject totheconditionthat c 1 =1and c 2 =10.Thefollowingsocalled\GoldenSection Method"isusedtoobtaintheoptimalvalueof t .TheGoldensectionmethodis describedasfollows: 82

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Tominimizeafunction g ( t )subjectto a t b wecanusesocalledGolden sectionmethodandthestepstousethealgorithmareasfollo ws: Step1 .Chooseanallowablenaltolerancelevel andassumetheinitialinterval wheretheminimumliesis[ a 1 ;b 1 ]=[ a;b ]andlet 1 = a 1 +(1 )( b 1 a 1 ) 1 = a 1 + ( b 1 a 1 ) Take =0 : 618,whichisapositiverootof c 2 + c 1=0,andevaluate g ( 1 )and g ( 1 ),letk=1andgotostep2. Step2 .If b k a k ,stopastheoptimalsolutionis t =( a k + b k ) = 2 : otherwise, if g ( k ) >g ( k ),gotostep3;andif g ( k ) g ( k ),gotostep4. Step3 :Let a k +1 = k and b k +1 = b k .Furthermorelet k +1 = k and k +1 = a k +1 + ( b k +1 a k +1 ).Evaluate g ( k +1 )andgotostep5. Step4 :Let a k +1 = a k and b k +1 = k .Furthermorelet k +1 = k and k +1 = a k +1 +(1 )( b k +1 a k +1 ).Evaluate g ( k +1 )andgotostep5. Step5 :Replace k by k +1andgotostep1. Toimplementthismethodtoourproblemweproceedasbelow Iteration1Consider[ a 1 ;b 1 ]=[0 ; 10], =0 : 618sothat1 =0 : 382 1 = a 1 +(1 )( b 1 a 1 )=3 : 82and 1 = a 1 + ( b 1 a 1 )=6 : 18 : ECU( 1 )=8 : 561andECU( 1 )=8 : 570. Since ECU( 1 ) ECU( 1 ) thenextintervalwheretheoptimalsolutionliesis[0 ; 6 : 18] 83

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Iteration2[ a 2 ;b 2 ]=[0 ; 6 : 18] 2 =2 : 36,and 2 =3 : 82 ECU( 2 )=8 : 533andECU( 2 )=8 : 561 Since ECU( 2 ) ECU( 2 ) thenextintervalwheretheoptimalsolutionliesis[0 ; 3 : 82] Iteration3[ a 3 ;b 3 ]=[0 ; 3 : 82] 3 =1 : 459and 3 =2 : 36 ECU( 3 )=8 : 516andECU( 3 )=8 : 533 Since ECU( 3 ) ECU( 3 ) thenextintervalwheretheoptimalsolutionliesis[0 ; 2 : 36] Iteration4[ a 4 ;b 4 ]=[0 ; 2 : 36] 4 =0 : 901and 3 =1 : 459 ECU( 4 )=8 : 613andECU( 3 )=8 : 516 Since ECU( 4 ) ECU( 4 ) thenextintervalwheretheoptimalsolutionliesis[0 : 901 ; 2 : 36] : 84

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Iteration5[ a 5 ;b 5 ]=[0 : 901 ; 2 : 36] 5 =1 : 459and 5 =1 : 803 ECU( 5 )=8 : 516andECU( 5 )=8 : 517 Since ECU( 5 ) ECU( 5 ) thenextintervalwheretheoptimalsolutionliesis[0 : 901 ; 1 : 803] Iteration6[ a 6 ;b 6 ]=[0 : 901 ; 1 : 803] 6 =1 : 246and 6 =1 : 459 ECU( 6 )=8 : 528andECU( 6 )=8 : 516 Since ECU( 6 ) ECU( 6 ) thenextintervalwheretheoptimalsolutionliesis[1 : 246 ; 1 : 803] Iteration7[ a 7 ;b 7 ]=[1 : 246 ; 1 : 803] 7 =1 : 459and 7 =1 : 590 ECU( 7 )=8 : 516andECU( 7 )=8 : 514 Since ECU( 4 ) ECU( 4 ) thenextintervalwheretheoptimalsolutionliesis[1 : 459 ; 1 : 803] Ifwexthe levellessthanorequalto0 : 5wecanconcludethattheoptimumvalue liesintheinterval[1 : 459 ; 1 : 803]anditisgivenby 1 : 459+1 : 803 2 =1 : 631. Wehaveperformthisnumericalexampleassumingthatthefai luredatafollows the TSL (1 ; 1)modelandweobtainthattooptimizethecostwehavetosche dulethe maintenancetimeafter1.631unitsoftime. 85

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7.3BlockReplacementPolicyandTSLProbabilityDistribut ion Hereweconsiderthecaseofthesocalled\Block-Replacemen tPolicy"ortheconstant intervalpolicy.Inthispolicyweperformpreventivemaint enanceonthesystemafterit hasbeenoperatingatotalof t unitesoftime,regardlessofthenumberofintervening failures.Incasethesystemhasfailedpriortothetime t ,minimalrepairwillbe performed.Weassumethattheminimalrepairwon'tchangeth efailurerateofthe systemandthepreventivemaintenancerenewsthesystemand itbecomeasgoodas new.Thus,wewanttondthe t thatminimizestheexpectedrepairandpreventive maintenancecost.Thetotalexpectedcostperunittimeforp reventivereplacement attime t ,denotedbyECU( t )isgivenby ECU( t )= Totalexpectedcostintheinterval(0 ;t ) Lengthoftheinterval : Thetotalexpectedcostintheinterval(0 ;t )equalstothecostofpreventativemaintenanceplusthecostoffailuremaintenance,thatis= c 1 + c 2 M ( t ),where M ( t )is theexpectednumberoffailureintheinterval(0 ;t ) Hence, ECU ( t )= c 1 + c 2 M ( t ) t : Butweknowthattheexpectednumberoffailureintheinterva l(0 ;t )istheintegral ofthefailureratefunction,thatis M ( t )= E ( N ( t ))= H ( t )= Z t 0 h ( t ) dt SoifthefailureofthesystemfollowstheTSLdistributionw eknowthat M ( t )= Z t 0 h ( t ) dt = (1+ ) t log((2+2 )exp( t = ) 1)+log(2 +1) Thuswehave ECU( t )= c 1 + c 2 [ (1+ ) t log((2+2 )exp( t = ) 1)+log(2 +1)] t : 86

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Againwewouldliketominimizethisequationsubjecttothec ondition c 1 =1and c 2 =10.Weshalluseagainsocalled"GoldenSectionMethod"too btainthevalue of t thatminimizesECU( t ) Iteration1consider[ a 1 ;b 1 ]=[0 ; 10], =0 : 618sothat1 =0 : 382 1 = a 1 +(1 )( b 1 a 1 )=3 : 82and 1 = a 1 + ( b 1 a 1 )=6 : 18 : ECU( 1 )=9 : 523andECU( 1 )=9 : 697. Since ECU( 1 ) ECU( 1 ) thenextintervalwheretheoptimalsolutionliesis[0 ; 6 : 18] Iteration2[ a 2 ;b 2 ]=[0 ; 6 : 18] 2 =2 : 36,and 2 =3 : 82 ECU( 2 )=9 : 30andECU( 2 )=9 : 523 Since ECU( 2 ) ECU( 2 ) thenextintervalwheretheoptimalsolutionliesis[0 ; 3 : 82] Iteration3[ a 3 ;b 3 ]=[0 ; 3 : 82] 3 =1 : 459and 3 =2 : 36 ECU( 3 )=9 : 124andECU( 3 )=9 : 30 Since ECU( 3 ) ECU( 3 ) thenextintervalwheretheoptimalsolutionliesis[0 ; 2 : 36] Iteration4[ a 4 ;b 4 ]=[0 ; 2 : 36] 4 =0 : 901and 3 =1 : 459 ECU( 4 )=9 : 102andECU( 3 )=9 : 124 87

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Since ECU( 4 ) ECU( 4 ) thenextintervalwheretheoptimalsolutionliesis[0 ; 1 : 459] Iteration5[ a 5 ;b 5 ]=[0 ; 1 : 459] 5 =0 : 557and 5 = : 901 ECU( 5 )=9 : 405andECU( 5 )=9 : 102 Since ECU( 5 ) ECU( 5 ) thenextintervalwheretheoptimalsolutionliesis[0 : 557 ; 1 : 459] Iteration6[ a 6 ;b 6 ]=[0 : 557 ; 1 : 459] 6 =0 : 9015and 6 =1 : 114 ECU( 6 )=9 : 102andECU( 6 )=9 : 08 Since ECU( 6 ) ECU( 6 ) thenextintervalwheretheoptimalsolutionliesis[ : 901 ; 1 : 459] Iteration7[ a 7 ;b 7 ]=[0 : 901 ; 1 : 459] 7 =1 : 114and 7 =1 : 245 ECU( 7 )=9 : 08andECU( 7 )=9 : 09 Since ECU( 4 ) ECU( 4 ) thenextintervalwheretheoptimalsolutionliesis[ : 901 ; 1 : 245] Againifwexthe levellessthanorequalto0 : 5wecanconcludethattheoptimum valueliesintheinterval[0 : 901 ; 1 : 245]anditisgivenby 0 : 901+1 : 245 2 =1 : 07. 88

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AsinthecaseofAgereplacementcaseinthisnumericalexamp leweassumethatthe failuredatafollowsthe TSL (1 ; 1)modelandwehaveseenthattooptimizethecost wehavetoschedulethemaintenancetimeevery1.07unitsoft ime. 7.4MaintenanceOveraFiniteTimeSpanTheproblemconcerningthepreventivemaintenanceoveran itetimespanisofgreat importantinindustry.Itcanbeviewedintwodierentprosp ective;whetherthetotal numberofreplacements(Failure+planned)timesareknowno rnot.Therstcaseis straightforwardanditisknownintheliteraturefromalong time.Barlowetal.(1967) derivetheexpressionforthiscase.Let T bethetotaltimespanwhichmeanswe wouldliketominimizethecostduetoreplacementorduetopl annedreplacement until T = T .Let C n ( T ;T )betheexpectedcostinthetimespan0to T ,[0 ;T ], consideringonlytherstnreplacementsfollowingapolicy ofreplacementatinterval T.Itisclearthatconsideringthecasewhen T T isequivalenttonoplanned replacement.Itisclearthat C 1 ( T ;T )= 8<: c 2 F ( T )if T T; c 2 F ( T )+ c 1 (1 F ( T ))if T T Thus,for n =1 ; 2 ; 3 ;::: ,wehave C n +1 ( T ;T )= 8>><>>: Z T 0 [ c 2 + C n ( T y;T )] dF ( y )if T T; Z T 0 [ c 2 + C n ( T y;T )] dF ( y )+ C ( T ;T )otherwise. (7.4.7) where C ( T ;T )=[ c 1 + C n ( T T;T )][1 F ( T )] : Herewewouldliketodevelopastatisticalmodelwhichcanbe usedtopredictthe totalcostbeforeweactuallyusedanyitem.LetTbetheprede terminedreplacement time.Wealwaysreplaceanitemexactlyatthetimeoffailure orThoursafter itsinstallation,whicheveroccursrst.Let denotesthersttimetofailureor 89

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replacementthenwehave E (cost)= Z T 0 [ c 2 + C T ( T y )] f ( y ) dy +[1 F ( T )][ c 1 + C T ( T T )] where c 1 isthecostforpreventivemaintenanceand c 2 ( >c 1 )isthecostforfailure maintenance. Thus,wecanwrite C ( T )= Z T 0 [ c 2 + C ( T y )] f ( y ) dy +[1 F ( T )][ c 1 + C ( T T )] = c 2 F ( T )+ c 1 R ( T )+ R ( T ) C ( T T )+ Z T 0 C ( T y )] f ( y ) dy; and C 0 ( T )=( c 2 c 1 ) F 0 ( T )+ R 0 ( T ) C ( T T ) R ( T ) C 0 ( T T ) + C ( T T ) f ( T ) =( c 2 c 1 ) F 0 ( T )+[ R 0 ( T )+ f ( T )] C ( T T ) R ( T ) C 0 ( T T ) Weneedtosolvethisdierentialequationtondthetotalco st.Wewouldlike toconsideranumericalexampletoseewhethertheminimumex istifweassume thefailuremodelbeing TSL (1 ; 1).Wegeneratearandomsampleofsize100from TSL (1 ; 1)andxatimeTtoperformpreventivemaintenance.Weconsi derthe preventivemaintenancecost c 1 =1andfailurereplacementcost c 2 =1 ; 2and10. Werepeattheprocessseveraltimesandcomputedthetotalco stforrst40failures andgotthetablebelow.Table7.1showstheexistenceofmini mumcost. 90

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T FC10 FC2 FC1 1 : 00 340 : 55 88 : 95 57 : 50 1 : 01 347 : 25 89 : 65 57 : 45 1 : 02 336 : 95 87 : 75 56 : 60 1 : 03 342 : 95 88 : 15 56 : 30 1 : 04 339 : 15 87 : 15 55 : 65 1 : 05 341 : 25 87 : 25 55 : 50 1 : 06 334 : 40 86 : 40 55 : 40 1 : 07 343 : 75 87 : 35 55 : 30 1 : 08 332 : 15 84 : 95 54 : 05 1 : 09 338 : 55 85 : 81 54 : 22 1 : 10 318 : 48 82 : 67 53 : 19 1 : 11 327 : 68 84 : 04 52 : 59 1 : 12 344 : 76 86 : 48 54 : 19 1 : 13 333 : 70 84 : 50 53 : 35 1 : 14 340 : 40 85 : 20 53 : 30 1 : 15 338 : 86 84 : 68 53 : 90 1 : 16 331 : 28 82 : 90 53 : 86 1 : 17 338 : 27 84 : 09 54 : 31 1 : 18 335 : 24 83 : 05 53 : 52 1 : 19 341 : 90 84 : 00 54 : 76 1 : 20 363 : 90 87 : 50 56 : 95 Table7.1:Comparisonsofcostsfordierentvaluesofpreve ntivemaintenancetimesInTable7.1, FC i ;i =1 ; 2&10representsthetotalcostduetopreventivemaintenancecost c 1 =1andthefailurereplacementcost c 2 = i;i =1 ; 2&10.Tableshows thattheminimum FC i existsaboutat T =1 : 1unitsoftime. 7.5ConclusionInthischapterwehavestudiedtheanalyticalbehaviorofth eTSLpdfwhenitisused tomodelpreventativemaintenancestrategiesinboththeag ereplacementandblock replacementpolices.Wealsohavedevelopedallessentiale stimatesoftheparameters thatareinheritedinsuchanalysis. Usingnumericaldatawehaveillustratedtheusefulnessofd eterminingtheoptimumcostutilizingapreventivemaintenancethatwasinitia llymodeledbyadierentialequation. 91

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Chapter8 FutureResearch InthisChapterweshallidentifysomeimportantresearchpr oblemsthatresultedfrom thepresentstudy,thatweshallinvestigateinthenearfutu re. Itshouldbenotedthatinthepresentstudywearerestricted toaunivariateskew Laplaceprobabilitydistribution.Itisofgreatinterestt oextendthisdistributionto themultivariatecases.Infact,arandomvector z =( Z 1 ;:::Z p ) T isap-dimensional skewLaplacerandomvariabledenotedby z SL p (n ; ) ,ifitiscontinuouswith pdfgivenby f ( z )=2 g p (z;n) G p ( T z) ; z 2 R p ; where g p (z;n) and G p ( T z) denotesthepdfandcdfofthep-dimensionalmultivariateLaplaceprobabilitydistributionwiththecorrelation matrix n and isthevector ofshapeparameters.Allanalyticaldevelopmentsinthepre sentstudy,webelieve, canbeextended,however,itmaybringssomediculties. Itshouldbealsonotedthatinthepresentstudyweintroduce dseveralrealworld datastrictlyforthepurposeofidentifyingthegoodnessof tofthedierenttypes ofprobabilitydensityfunctionsthatwehaveintroduced.F urthermorewecompared thetnesswiththetnessoftheactualprobabilitydensity thatwasusedtoanalyze thesubjectdata.Wehavedemonstratedthatthedevelopedpr obabilitydistribution givesabetterprobabilisticcharacterizationonthisreal worldphenomenon. 92

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Thus,itistheaimofthefutureresearchprojectstostatist icallyfullyanalyzeand modelthisrealworlddatausingtheproposedanalyticalres ultswehavedeveloped. Weanticipatethatouranalysiswillresultinbetterdecisi onsandestimationof thevariousunknownsrelatedtoeachoftheprojects,becaus eouranalyticalmethods gavebettertsthantheonewhichwereusedtoanalyzethesub jectdata. Wehavepresentedtworealworlddatanamelythecurrencyexc hangedataandthe pressurevesselsdatainChapter4andChapter6respectivel y.Inthecurrencyexchangedataweweremainlyinterestedonwhetherourpropose dmodels,theskew Laplaceprobabilitydistribution,tstheexchangerateda tabetterthanthetraditionallyusedGaussianmodel.Wehaveobservedgraphically andstatisticallythat thegoodnessoftoftheSLpdftsmuchbetter.Nowweareinte restedtostudythe possibleimpactofchoosingthismodelonthenancialanaly sisanddecisionmaking ofthisdata.Morespecically,weareinterestedonestimat ionandtheinferential statisticalstudyofthedata. Weshallalsoinvestigateanalyticallyandbysimulationth erelationshipbetween theskewLaplaceprobabilitydistributionandtheskewNorm alprobabilitydistribution.Wewillusethiscurrencyexchangeratedatatoverifyt herelation. InChapter6wehaveobserved,graphicallyandstatisticall ythatthebygoodness oftofthetruncatedskewLaplace(TSL)probabilitydistri butionttedthepressure vesselsfailuredatabetterthanthetwoparametergammadis tributionthatwasused toanalyzethedata.Nowitisofinteresttoinvestigatethei nferentialstudyofthe TSLmodeltothisdataandcomparethendings. Finally,inChapter7wehaveobservedthattheTSLprobabili tydistributioncan beusedinthepreventivemaintenanceoveraninniteandove ranitetimespan.We shallstudytheexistence,instabilitybehaviorofthedela ydierentialequationthat wasresultedoncomputingtheexpectedcostoveranitetime span.Furthermore, 93

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weshallstudythemostappropriatenumericaltechniquetoo btaintheestimatesof thesolutionofthenonlineardierential,delayequationa ndapplyquasilinearization methodologytoreducethesubjectdierentialsystemintoa quasi-linearformsothat wecanobtainanexactanalyticalsolutionofthesystem.The setwoapproacheswill becomparedtodeterminetheireectiveness. 94

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References [1]Aryal,G.,Rao,A.N.V.(2005) Reliabilitymodelusingtruncatedskew-Laplace distribution, NonlinearAnalysis,Elsevier,63,e639-e646 [2]Aryal,G.,Nadarajah,S.(2005) OntheskewLaplacedistribution ,Journalof informationandoptimizationsciences,26,205-217. [3]Andrews,D.F.,Bickel,P.J.,Hampel,F.R.,Huber,P.J., Rogers,W.H.and Tukey,J.W.(1972). RobustEstimatesofLocation ,PrincetonUniversityPress. Princeton,NewJersey. [4]Azzalini,A.(1985) Aclassofdistributionsthatincludesthenormalones ,Scand. J.Statistics,12,171-178. [5]BarlowR.E.andF.Proschen(1962) StudiedinAppliedProbabilityandmanagementScience, StanfordUniversityPress,Stanford,California. [6]Cox,D.R.(1961),Testsofseparatefamiliesofhypothes es, Proceedingsofthe FourthBerkelySymposiuminMathematicalStatisticsandPr obability, Berkely, UniversityofCaliforniaPress,105-123 [7]Edgeworth,F.Y.(1886), Thelawoferrorandtheeliminationofchance ,PhilosophicalMagazine.21,308-324. [8]Genton,M.G.,(2004) Skew-EllipticalDistributionsandtheirApplications Champman & Hall/CRC,133{140. [9]Gupta,A.K.,Chang,F.C.andHuang,W.J.(2002), Someskew-symmetric models.RandomOperatorsandStochasticEquations. 10,133{140. [10]Hoaglin,D.C.,Mosteller,F.andTukey,J.W.(eds)(198 3) UnderstandingRobust andExploratoryDataAnalysis ,JohnWileyandSons.NewYork. 95

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[11]Ihaka,R.andGentleman,R.(1996) R:Alanguagefordataanalysisandgraphics. JournalofComputationalandGraphicalStatistics.5,299{ 314. [12]Julia,OandVives-RegoJ.(2005), Skew-LaplacedistributioninGram-negative bacterialaxeniccultures:newinsightsintointrinsiccel lularheterogeneity. Microbiology,151,749-755 [13]Keating,J.P.,Glaser,R.E.,andKetchum,N.S.(1990) TestingHypothesisabout theshapeparameterofaGammaDistribution. Technometrics,32,67-82. [14]Klein,G.E.(1993) Thesensitivityofcash-rowanalysistothechoiceofstatis tical modelforinterestratechanges ,TransactionsoftheSocietyofActuariesXLV, 79-186. [15]Kotz,S.Kozubowski,T.J.,Podg o rski,K.(2001), TheLaplaceDistributionand Generalization. ,Birkh a user. [16]Kozubowski,T.J.,Podg o rski,K.(2000), AssymetricLaplacedistributions Math.Sci.25,37-46 [17]Kundu,D.(2004).DiscriminatingbetweentheNormalan dtheLaplacedistributions, Report ,DepartmentofMathematics,IndianInstituteofTechnolog y Kanpur,India. [18]Leadbetter,M.R.,Lindgren,G.andRootzen,H.(1987) ExtremesandRelated PropertiesofRandomSequencesandProcesses ,SpringerVerlag.NewYork. [19]McGill,W.J(1962) RandomFluctuationsofresponserate ,Psychometrika,27,317 [20]Pal,N.,Jin,C.,Lim,W.K,(2006) HandbookofExponentialandrelateddistributionsforEngineersandScientists ,Chapman & Hall/CRC [21]Poiraud-Casanova,S.,Thomas-Agnana,C.(2000) Aboutmonotoneregression quantiles ,Statist.Probab.Lett.48101-104 96

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[22]Polya,G.andSzego,G.(1976). ProblemsandTheoremsinAnalysis (volumeI), Springer-Verlag.NewYork. [23]Rao,M.,Chen,Y.,Vemuri,B.C.andWang,F.(2003).Cumu lativeresidual entropy,anewmeasureofinformation. Report ,DepartmentofMathematics, UniversityofFlorida,Gainsville,Florida,USA. [24]Renyi,A.(1961).Onmeasuresofentropyandinformati on, Proceedingsofthe 4thBerkeleySymposiumonMathematicalStatisticsandProb ability ,Vol.I,pp. 547{561,UniversityofCaliforniaPress.Berkeley. [25]Trivedi,K.S.(1982).ProbabilityandStatisticswith Reliability,Queuingand ComputerScienceApplications Prentice-HallInc. 97

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AbouttheAuthor TheauthorwasborninNepal.HereceivedB.Sc.(inMathemati cs,Physicsand Statistics)andM.Sc.(inMathematics)fromTribhuvanUniv ersity,Kathmandu, Nepal.In2000,hewasawardedtheUNESCOscholarshiptostud yinICTP(InternationalCentreforTheoreticalPhysics)Trieste,Italy.Heh asearnedICTPDiplomain Mathematicsin2001.HehasalsoearnedM.A.inMathematicsf romtheUniversity ofSouthFloridain2003.Hehasbeenateachingassistantatt heDepartmentof MathematicssinceFall2001.Hewasawardedwitharecogniti onoftheProvost's awardforoutstandingteachingbyagraduateteachingassis tantin2006.