Application of Mml Methodology

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HacettepeJournalofMathematicsandStatisticsVolume39 (3)(2010),449 460APPLICATIONOFMMLMETHODOLOGYTOANSERIESPROCESSWITHWEIBULLDISTRIBUTIONHalilAydogdu,BirdalSenogluandMahmutKaraReceived21 : 12 : 2009 : Accepted12 : 03 : 2010AbstractIn an -series process, explicit estimators of the parameters , and 2are obtained by using the methodology of modied maximum likelihood(MML)whenthedistributionof therstoccurrencetimeof aneventisassumedtobeWeibull. MonteCarlosimulationsareperformedtocompare the eciencies of the MML estimatorswith the correspondingnonparametric(NP)estimators. WealsoapplytheMMLmethodol-ogytotworeal lifedatasets toshowtheperformance of theMMLestimatorscomparedtotheNPestimators.Keywords: -series process, Modiedlikelihood, Nonparametric, Eciency, MonteCarlosimulation.2000AMSClassication: 60 G55,62 F10,62 G08.1. IntroductionIn the statistical literature, a counting process (CP) {N(t), t 0} is a common methodof modelingthetotal numberof eventsthathaveoccurredintheinterval (0, t]. If thedataconsistofindependent andidenticallydistributed(iid)successiveinterarrivaltimes(i.e.,there isno trend),arenewalprocess(RP) canbeused. However,this isnot alwaysthe casebecauseitismorereasonableto assumethat the successiveoperatingtimeswillfollowamonotonetrendduetotheageingeectandaccumulatedwear,seeChanetal.[8]. UsinganonhomogeneousPoissonprocesswithamonotoneintensityfunctionisoneapproach to modeling these trends, see Cox and Lewis [9] and Ascher and Feingold [2]. Amoredirectapproachistoapplyamonotonecountingprocessmodel, see, forexample,Lam[13, 14], LamandChan[15], andChanet al. [8]. Braunet al. [6] denedsuchamonotoneprocessasbelow.Ankara University, Department of Statistics, 06100 Tandogan, Ankara, Turkey. E-mail:(H. Aydogdu) aydogdu@science.ankara.edu.tr (B. Senoglu) senoglu@science.ankara.edu.tr(M.Kara)mkara@science.ankara.edu.trCorrespondingAuthor.450 H. Aydogdu,B. Senoglu,M.Kara1.1.Denition. LetXkbethetimebetweenthe(k 1)thandktheventofacountingprocess {N(t), t 0} for k =1, 2, . . .. The counting process {N(t), t 0} is saidtobean-series process withparameter if thereexists areal number suchthatkXk,k= 1, 2, . . .areiidrandomvariableswithadistributionfunctionF.The-seriesprocesswasrstintroducedbyBraunetal. [6]asapossiblealternativeto the geometric process (GP) in situations where the GP is inappropriate. They appliedit to some reliability and scheduling problems. Some theoretical properties of the -seriesprocess are giveninBraunetal. [6,7]. Clearly,the Xks form a stochasticallyincreasingsequencewhen< 0; theyarestochasticallydecreasingwhen >0. When= 0, alloftheXkareidenticallydistributedandan-seriesprocessreducestoaRP.If F is anexponential distributionfunctionand=1, thenthe-series process{N(t), t 0}isalinearbirthprocess[7].Assumethat thedistributionfunctionFfor an-series process haspositivemean(F(0) < 1),andnitevariance2. Then(1.1) E(Xk) =kandVar(Xk) =2k2, k = 1, 2, . . ..Thus, , and2arethemostimportantparametersfor an-series processbecausetheseparameterscompletelydeterminethemeanandvarianceofXk.Inthis paper, we studythestatistical inference problemfor -series process withWeibulldistributionandobtainthe explicitestimatorsoftheparameters, and2byadopting the method of modiedlikelihood. We recallthat the mean and variance of theWeibulldistributionaregivenby(1.2) = _1 +1a_band2=__1 +2a_2_1 +1a__b2,respectively. Here, a and b are the shape and scale parameters of the Weibull distribution,respectively.Sinceobtainingexplicitestimatorsoftheparametersissimilartothatgivenin[25],wejustreproducetheestimatorsandomitthedetails, seealso[10], [11] and[23]. Themotivation for this paper comes from the fact that the Weibull distribution is not easy toincorporateintothe-seriesprocessmodelbecauseofdicultiesencounteredinsolvingthe likelihoodequations numerically. To the best ofour knowledge,this is the rst studyapplyingtheMMLmethodologytoan-seriesprocesswhenthedistributionoftherstoccurrencetimeisWeibull.Theremainder of thispaper is organizedas follows: Likelihoodequations for esti-matingtheunknownparameters andthecorrespondingMMLestimators aregiveninSection2. Section3presentstheelementsoftheinverseoftheFisherinformationma-trix. Section4presentsthesimulationresultsforcomparingtheecienciesoftheMMLestimatorswiththeNPestimators. Tworeal lifeexamplesaregiveninSection5. Con-cludingremarksaregiveninSection6.2. LikelihoodequationsAs mentionedinSection1, Yk=kXk, k =1, 2, . . . , nare iidWeibull randomvariables. Thelikelihoodequations, ln L= 0, ln La= 0and lnLb= 0areobtainedbytakingtherstderivativesofthelog-likelihoodfunctionwithrespecttotheparameters, aandb. Forexample,weseethatthelikelihoodequationfortheparameteris ln L=n

k=1ln k n

k=1_kXkb_ln k = 0.ApplicationofMMLMethodology 451The solutions of the likelihood equations give us the ML estimators of the parameters , aand b. However, the ML estimatorscannot be obtainedexplicitly or numericallybecausetherstderivativesof thelikelihoodfunctioninvolvepowerfunctionsof theparameter,aswellastheshapeparametera.Toovercomethesediculties,wetakethelogarithmoftheYksasshownbelow:(2.1) ln Yk= ln k + ln Xk, k= 1, 2, . . . , n.Itisknownthattheln Yksareiidextremevalue(EV)randomvariableswiththeprob-abilitydensityfunctiongivenby(2.2) f(w) =1exp_w _exp_exp_w __, w R; > 0, R,where= ln bisthelocationparameterand= 1/aisthescaleparameter.We rst obtain the estimatorsof the unknown parameters in the extreme value distri-bution,andthenobtaintheestimatorsoftheWeibullparametersbyusingthefollowinginversetransformations:(2.3) a = 1/andb = exp().ThelikelihoodfunctionforlnXk,k = 1, 2, . . . , nisfoundbyusing(2.1)and(2.2):(2.4) L(, , ) =1nexp_n

k=1lnXk +ln k exp_ln Xk +ln k __.Forthesakeofsimplicity,weusethenotationsckinsteadofln kin(2.4).Then, it is obvious that maximizing L(, , ) with respect to the unknown parameters, andisequivalenttoestimatingtheparametersin(2.5),(2.5) ln xk= ck +k(1 k n),whenk EV(0, ).Thenthelikelihoodequationsaregivenby(2.6) ln L=1n

k=1ck1n

k=1g(zk)ck= 0 ln L=1n

k=1g(zk) n= 0and ln L=1n

k=1g(zk)zk1n

k=1zkn= 0,wherezk=k, (k = 1, 2, . . . , n)andg(z) = exp(z).TheMLestimatorsofthe extremevaluedistributionparametersarethe solutionsoftheequationsin(2.6). Becauseof theawkwardfunctiong(z), itisnoteasytosolvetheseequations. TheMLestimates are, therefore, elusive. Sinceit is impossibletoobtainestimatorsoftheunknownparametersinaclosedform, weresorttoiterativemethods.Howeverthatcanbeproblematicforreasonsofi) multipleroots,ii) non-convergenceofiterations,oriii) convergencetowrongvalues;452 H. Aydogdu,B. Senoglu,M.Karasee, Barnett [4] andVaughan[24]. If thedatacontainoutliers, theiterations for thelikelihoodequationsmightneverconverge,seePuthenpuraandSinha[17].To overcome these diculties,we use the method of modiedmaximum likelihoodin-troduced by Tiku [19,20], see also Tikuetal. [22]. The method linearizes the intractableterms in(2.6)andcallsthe resultingequations the modiedlikelihoodequations. ThesolutionsoftheseequationsarethefollowingclosedformMMLestimators(see[10, 11,23]and[25]):(2.7) = D E, = ln x[.] + c[.] +m and =B +B2+ 4nC2_n(n 2)(biascorrected)whereD=n

k=1(1 ak)(c[k] c[.])n

k=1bk(c[k] c[.])2, E=n

k=1bk(ln x[k]ln x[.])(c[k] c[.])n

k=1bk(c[k] c[.])2,ln x[.]=n

k=1bk ln x[k]m, c[.]=n

k=1bkc[k]m, =n

k=1(ak1), m =n

k=1bk,B=n

k=1(ak1)_(lnx[k]ln x[.]) E(c[k] c[.])andC=n

k=1bk_(lnx[k]ln x[.]) E(c[k] c[.])2.Then,by(1.2)and(2.3),theMMLestimatorsofand2areobtainedas(2.8) = (1 + ) exp()and 2=_(1 + 2 ) 2(1 + )exp(2)respectively.It should be noted that the divisor nin the denominator of was replaced by_n(n 2)asabiascorrection. SeeVaughanandTiku[25] andTikuandAkkaya[23]fortheasymptoticandsmallsamplespropertiesoftheMMLestimators.3. TheFisherinformationmatrixTheelementsoftheinverseoftheFisherinformationmatrix(I1)aregivenbyV11= 2__n

k=1c2k_n

k=1ck_2_n_,V12= 2n

k=1ck__nn

k=1c2k_n

k=1ck_2_, V13= 0,V22= 2n

k=1c2k__nn

k=1c2k_n

k=1ck_2_+ (1 )2/n2,V23= 62(1 )/n2andV33= 62/n2,whereistheEulerconstant.It shouldbe notedthat the Fisher informationmatrix I is symmetric, so V21 =V12, V31 =V13 andV32 =V23. Thediagonal elements inI1providetheasymptoticvariances of the ML estimators, i.e., V ( ), V () and V ( ). These variances are also knownas the minimum variance bounds (MVBs) for estimating , and . The MML estimatorsApplicationofMMLMethodology 453areasymptoticallyequivalent totheMLestimatorswhenthe regularityconditionshold,see Bhattacharrya[5] andVaughanandTiku[25]. Therefore, we canconclude thattheMMLestimatorsarefullyecient, i.e., theyareasymptoticallyunbiasedandtheircovariancematrixisasymptoticallythesameastheinverseof I. SeeTable1for thesimulatedvariancesoftheMMLestimatorsin(2.7)andthecorrespondingMVBvalues.Table1. ThesimulatedvariancesoftheMMLestimatorsandthecorrespondingMVBvaluesSimulatedvariances MVBn 30 0.0132 0.0908 0.0054 0.0119 0.0828 0.005150 0.0069 0.0650 0.0031 0.0065 0.0623 0.0030100 0.0030 0.0424 0.0015 0.0029 0.0415 0.0015Itis clear fromTable1that thesimulatedvariances of theMMLestimators andthecorrespondingMVBvaluesbecomecloseasnincreases. Therefore,theMMLestimatorsarehighlyecientestimators.4. SimulationresultsAnextensiveMonteCarlosimulationstudywascarriedoutinordertocomparetheecienciesofthe MMLestimatorsandthe NP estimatorsgivenbelow;seeAydogdu andKara[3].(4.1) =n

k=1ln kn

k=1ln Xknn

k=1ln Xk ln knn

k=1(ln k)2_n

k=1ln k_2, 2= exp(2) 2eand =___rootof2 2ln 2 2 2= 0, < 0n

k=1k Xk/n

k=1k2 , 0 < 0.7n

k=1Xk/n

k=1k , > 0.7where=n

k=1lnkn

k=1ln Xk ln k n

k=1(ln k)2n

k=1ln Xk_n

k=1lnk_2nn

k=1(ln k)2and 2e=_n

k=1(ln Xk)21n_n

k=1ln Xk_2_ 2_n

k=1(lnk)21n_n

k=1ln k_2_n 2.All thecomputationswereconductedinMATLAB7. T