Comparing of some estimation methods for …

1 Hacettep eJournalofMathematicsandStatisticsVolume 45(5)(2016),1605 1620 Comparingofsomeestimationmetho dsforparametersoftheMarshall-Olkingenera lizedexp onentialdistributionunderprogressiveTyp e-IintervalcensoringHamzehTorabi ,MasumehForo oghi andHosseinNadeb AbstractInthispap er,weestimatetheparametersoftheMarshall- Olkingener-alizedexp onentialdistributionunderprogressiveTyp e-Iintervalcen-soringbasedonmaximumlikel iho o d,momentmetho ,theseestimatemeth-o :EMalgorithm,Generalizedexp onentialdistribution,Maximumlikeliho o destimate,Metho dofmoments,Typ cation:65C05,62N01, ductionAggarwala[2001]intro ducedTyp e-Iintervalandprogressivecensoringanddev elop edthestatisticalinferencefortheexp onentialdistributionbasedonprogressively Typ [2009]intro ducedtheconceptofprogressiveTyp e-IintervalcensoringtotheWeibulldistribu tionandcomparedmanydi erentestimationmetho dsfortwoparametersintheWeibulldistributi onviasimulation.

2 Corresp ondingAuthor. onential(GE)distributionhasthefollowingp robabilitydensityfunction(p df )f(t; , ) = (1 e t) 1e t,wheret >0, >0and > :F(t; , ) = (1 e t) ,andh(t; , ) = (1 e t) 1e t1 (1 e t) ,wheret >0;TheGEdistributionwasintro ducedbyGuptaandKunda[2001].Recently,Chen andLio[2010]intro ducedtheconceptofprogressiveTyp e-Iintervalcensoringforthegeneralizedexp onentialdistributionandcomparedmanydi erentestimationmetho onential(MOGE)distributionwas rstprop osedbyMarshallandOlkin[1997]andextensive lydiscussedbyAliceandJose[1999].ThePDFof theMOGE distributionwiththeparameters and is( )f(t; , ) = e t(1 (1 )e t)2, t >0,0< 1, > ,thedistributionfunctionandthehazardrate functionoftheMOGE distributionareasfollows:( )F(t; , ) =1 e t1 (1 )e t,andh(t; , ) = 1 (1 )e t,wheret > = 1,theMOGE distributionreducestotheconventionalexp ,distributionfunctionsandhazardratefunct ionsfordi erentvaluesof and aregivenin gures1,2and3,resp rsttwomomentsandvarianceoftheMOGE distributionaregivenbyE[T] = log ( )( 1) ,( )E[T2] =2 PolyLog[2,1 ](1 ) 2,( )V ar[T] = ( log ( )2+ 2( 1)PolyLog[2,1 ])( 1)2 2,wherePolyLog[2,1 ] = k=1(1 ) er,westudythemaximumlikeliho o destimates,estimatesviamomentmetho dsandestimatesviaprobabilityplotfortwopa rametersoftheMOGE distribu-tionundertheprogressiveTyp ducestheprogressiveTyp ,somemetho ,asimulationstudyisconductedtocom-pareth ep erformancesoftheseestimationmetho (t) = = =1 = (t) =3 = = = erentvaluesof and (MSE)

3 , e-IintervalcensoreddataSupp osethatnitemsareplacedonalifetestingprob lemsimultaneouslyattimet0= 0underinsp ectionatmpre-sp eci edtimest1< t2< .. < tmwheretmisthescheduledtimetoterminateth eexp ectiontime,ti,thenumb er,Xi,offailureswithin(ti 1,ti]isrecordedandRisurvivingitemsareran domlyremovedfromthelifetesting,fori= 1,.., erofsurvivingitemsatthetimetiisYi=n ij=1Xj i 1j= (t) = = =1 = (t) =3 = = = erentvaluesof and numb erofitemswithdrawnshouldnotb egreaterthanYiattimescheduleti,Ricouldb edeterminedbythepre-sp eci edp ercentageoftheremainingsurvivingunitsatt iforgiveni= 1,2,.., ,givenpre-sp eci edp ercentagevalues,p1,..,pm 1andpm= 1,forwithdrawingatt1< t2< .. < tm,resp ectively,Ri=bpiyicateachinsp ectiontimetiwherei= 1,2,.., ,aprogressivelyTyp e-Iintervalcensoredsamplecanb edenotedas(Xi,Ri,ti),i= 1,2,..m,wheresamplesizeisn=m i=1(Xi+Ri).)

4 NotethatifRi= 0,i= 1,2,..,m 1,thentheprogressivelyTyp e-IintervalcensoredsampleisaTyp e-Iintervalcensoredsample,X1,X2,..,Xm,Xm +1= e-Iintervalcensoredsampleb ecollectedasdescrib edab ove,b eginningwitharandomsampleofnunitswithaco ntinuouslifetimedistribution160901234501 234th(t) = = =1 = (t) =3 = = = erentvaluesof and functionF(.; ).Then,basedontheobserveddata,thelikelih o o dfunctionwillb easfollows:L( ) m i=1[F(ti; ) F(ti 1; )]Xi[1 F(ti; )] dsInthissection,wegivesomeestimationmeth o o oseaprogressiveTyp ( ),thelikeliho o dfunctionisL( , ) m i=1[1 e ti1 (1 )e ti 1 e ti 11 (1 )e ti 1]Xi[ e ti1 (1 )e ti]Ri,andthelog-likeliho o dfunctionis`( , ) m i=1 Xilog[1 e ti1 (1 )e ti 1 e ti 11 (1 )e ti 1]+m i=1 Rilog[ e ti1 (1 )e ti].Hence,wehavethefollowinglog-likeliho o dequations:( ) `( , ) = 0, `( , ) = and cannotb eobtainedinaclosedformbysolvingequations ( )andtheymustb ecalculatedusinganumericalmetho ,amid-p ointapproximationandtheEMalgorithmareint ro ducedasfollowsfor ndingtheMLEsof and.

5 Ointestimatorsbasedonprogres-sivelyTyp e-Iintervalcensoringcanb eobtainedbyassumingthatXifailureso ccurredatthecenteroftheinterval,mi=ti 1+ti2, o dfunctionfromtheMOGE distributioncanb esp eci edasfollows:log(L ) m i=1[Xilog(f(mi; , )) +Rilog(1 F(ti; , ))]=nlog + log m i=1Xi m i=1(Ximi+Riti) 2m i=1 Xilog(1 (1 )e mi) m i=1 Rilog(1 (1 )e ti).Therefore,themaximumlikeliho o destimateof , ,andthemaximumlikeliho o desti-mateof , ,arethesolutionofthesequelequations:( )n = 2m i=1 Xie mi1 (1 )e mi+m i=1 Rie ti1 (1 )e ti,andm i=1Xi =m i=1(Ximi+Riti) + 2(1 )m i=1 Ximie mi1 (1 )e mi+(1 )m i=1 Ritie ti1 (1 )e ti.( )1611 Thereisnoclosedformforthesolutionsof( )and( ),thusaniterativenumericalmetho disneededtoobtaintheparameterestimates, , and . o destimatoroftheparametersandusefulinavar ietyofincomplete-dataproblemswherealgori thmssuchastheNewton-Raphsonmetho dmaysometimesb ,therearetwostepscalledE-stepandtheM-ste p:Letyij,j= 1,2.

6 ,Xi,b ethesurvivaltimeswithinsubinterval(ti 1,ti]andzij,j=1,2,..,Ri,b ethesurvivaltimesforthosewithdrawnitemsa ttifori= 1,2,3,..,m,thenthelog-likeliho o d,log(L ),forthecompletelifetimesofnitemsfromthe MOGE distributionisgivenasfollows:log(L ) m i=1[Xi j=1log(f(yij, )) +Ri j=1log(f(zij, ))]=n(log + log ) m i=1[Xi j=1yij+Ri j=1zij] 2m i=1[Xi j=1log(1 (1 )e yij) +Ri j=1log(1 (1 )e zij)].( )Takingthederivativewithresp ectiveto and ,resp ectively,on( ),likeliho o dequationsareobtainedbyn = 2m i=1[Xi j=1e yij(1 (1 )e yij)+Ri j=1e zij(1 (1 )e zij)],andn = 2m i=1[Xi j=1(1 )yije yij(1 (1 )e yij)+Ri j=1(1 )zije zij(1 (1 )e zij)]+m i=1[Xi j=1yij+Ri j=1zij].TheEM- and ,say (0)and (0); ,theE-steprequirestocomputeE1i=E (k), (k)[Y Y [ti 1,ti)],E2i=E (k), (k)[Y Y [ti, )],E3i=E (k), (k)[e (k)Y1 (1 (k))e (k)Y Y [ti 1,ti)],E4i=E (k), (k)[e (k)Y1 (1 (k))e (k)Y Y [ti 1, )],E5i=E (k), (k)[Y e (k)Y1 (1 (k))e (k)Y Y [ti 1,ti)],1612andE6i=E (k), (k)[Y e (k)Y1 (1 (k))e (k)Y Y [ti 1, )],whereYisarandomvariablewhichhastheMOG E distributiondensityfunction( ).]]]]]

7 O o dequationsforcompletedata,wecanobtainthe estimates (k+1)=n2m i=1[Xi j=1E3i+Ri j=1E4i],and (k+1)=nm i=1[Xi j=1E1i+ 2(1 (k+1))E5i+Ri j=1E2i+ 2(1 (k+1))E6i]; + 1,theMLEsof and canb eobtainedbyrep eatingtheE-stepandM-stepuntilconvergence o dsarerequiredtocomputetheab ovecondi-tionalexp earandomvariablewhichhastheMOGEdis-tribu tiondensityfunction( ).Thekthmomentofadoublytruncatedgenerali zedexp onentialdistributionintheinterval(a,b)wh ere0< a < bisgivenbyE , [Yk Y [a,b)]=b aykf(y; , )dyF(b; , ) F(a; , )Equatingthesamplemomenttothecorresp ondingp opulationmomentuptothesecondorder,thefol lowingequationscanb eusedto ndtheestimatesofmomentmetho d:E[Y] =1n[m i=1 XiE , [Y|Y [ti 1,ti)] +RiE , [Y|Y [ti 1, )]],andE[Y2] =1n[m i=1 XiE , [Y2 Y [ti 1,ti)]]+[m i=1 RiE , [Y2 Y [ti 1, )]].Aniterativepro cedurecanb eemployedtosolvetheab oveequationsfor and and ,say (0)and (0)withk= 0; + 1thiteration, wecomputeE (k), (k)[Y|Y [ti 1,ti)]andE (k), (k)[Y2 Y [ti 1,ti)]andsolvethefollowingequationfor ,say (k+1):P( ) =[m i=1 XiE , [Y|Y [ti 1,ti)] +RiE , [Y|Y [ti 1, )]]2nm i=1[XiE , [Y2|Y [ti 1,ti)] +RiE , [Y2|Y [ti 1, )]],1613whereusing( )and( ),P( ) =E2[Y]/E[Y2] = log2 2(1 )PolyLog[2,1 ]).]]]]]]]]]]]

8 Thesolutionfor ,say (k+1),isobtainedthroughthefollowingequat ion: (k+1)log ( (k+1))( (k+1) 1) (k+1)=1n[m i=1 XiE (k), (k)[Y|Y [ti 1,ti)]+m i=1 RiE (k), (k)[Y|Y [ti 1, )]]; ,iftheconvergenceo ccursthenthecurrent (k+1)and (k+1)aretheestimatesof and bythemetho dofmoments;otherwisesetk=k+ e-Iintervalcen-soreddata,(Xi,Ri,ti),i= 1,2,..,m,ofsizen,thedistributionfunction attimeticanb eestimatedas F(ti) = 1 i j=1(1 pj), i= 1,2,..,m,where pj=Xjn j 1 k=0Xk j 1 k=0Rk, j= 1,2,.., ( ),wehavet= 1 log1 F(t)1 (1 )F(t).If F(ti)istheestimateofF(ti),thentheestimat esof and intheMOGE distributionbasedonprobabilityplotcanb eobtainedbyminimizingm i=1[ti+1 log1 F(ti)1 (1 ) F(ti)]2withresp ectto and . ,wegiveAshortalgorithmforsimulatingX1,X2 ,..,Xmfromarandomsampleofsizenputonlifet estattime0isthereforegivenb 0;Weusethefactthatfori= 1,..,m,Xi|Xi 1,..,X0,Ri 1.]]

9 ,R0 Binom(n i 1 j=1(Xj+Rj),F(ti) F(ti 1)1 F(ti 1)),andRi= pi(n i 1 j=1(Xi+Ri) Xi) . 0andletxsum=rsum= 0; ; + 1,exitthealgorithm; (n xsum rsum)1614andF(ti) F(ti 1)1 F(ti 1); pi(n i 1 j=1(Xi+Ri) Xi) orRobsi= min(n xsum rsum Xi,Ri),dep endingup onhowthecensoringschemeischosen; +Xi,rsum=rsum+Robsi; e-Iin-tervalcensoringundertheMOGE distributionlifetimemo dels,letusconsideranumer-icalexample, ,t2= ,t3= ,t4= ,t5= ,t6= ,t7= ,t8= ewithparameters( , ) = ( ,.06), erformancesoftheestimationpro ceduresdevelop edinthispap er,weconsiderthefollowingfourprogressive intervalcensoringschemeswhicharesimilart othepatternsofsimulationschemesusedinAg- garwala(2001)andalsousedinNgandWang(2009 )andChenandLio(2010):p(1)= (.25,.25,.25,.25,.5,.5,.5,.5,1),p(2)= (.5,.5,.5,.5,.25,.25,.25,.25,1),p(3)= (0,0,0,0,0,0,0,0,1),p(4)= (.25,0,0,0,0,0,0,0,1),wherecensoringinp( 1)islighterforthe (2).

10 P(3)istheconventionalintervalcensoringwh erenoremovalspriortotheexp erimentterminationandthecensoringinp(4)o nlyo and foriterativeprogressesofMLE,mid-p ointapproximation,EMalgorithm,momentmeth o dandprobabilityplotaregiventhesamevalues ,whichforeachsimulationrun, erformancesamongthefourcensoringschemes, thethirdschemep(3)providesthemostprecise resultsasseenfrom Bias , SD ( )and MSE ( )showninTable1andTable2fromthedisp ersionsoftheb oxplotsshownintheFigures1and2,thenfollow edbytheschemesp(4),p(1)andp(2). dellingtheMOGE distribution;Thisdatasetisexploredfrom[4 ] ,the rstcolumnshows7pre-assignedtimeintervals inyearswhichweredeterminedb eforetheexp eriment, ,[ti 1,ti),i=1,.., erofpatientswhoarediedinthetimeintervals , ,X1,..,X7and nally,thelastcolumnisthenumb erofpatientswhoweredropp edoutfromthestudyattherightendofeachtime interval;Thesedropp edpatientsareknowntob ,thelastcolumninTable3providesthevalueso fRi,i= 1.]