Semi-parametric Inference for Copula Models. for Dependently Truncated Data SUMMARY

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1 Sem-paramer feree for opla Models for Depedel Traed Daa TAKESH EMA WEJNG WANG ad H-NEN HNG Dvso of Bosass Deparme of Pharmaeal Sees Kasao vers Japa se of Sass Naoal hao-tg vers Hs-h Tawa O SMMA hs arle we vesgae he depede relaoshp bewee wo falre me varables whh rae eah oher haeb e al 6 proposed a sem-paramer model der he so-alled sem-srvval Arhmedea-opla assmpo ad dsssed esmao of he assoao parameer he rao probabl ad he margal fos Here he same model assmpo s adoped b dffere feree approahes are proposed For esmao of he assoao parameer we eed lao s odoal lkelhood approah 978 ad Wag s wo-b-wo able approah 3 o depede rao daa ad derve a relaoshp bewee he proposed mehods ad he mehod of haeb e al 6 based o -sass For margal esmao we propose a ovel algorhm ad derve a epl formla o oba he esmaor arge sample properes are esablshed o he foal dela mehod whh a hadles more geeral esmag fos ha he -sas approah Smlaos are performed ad he proposed mehods are appled o he rasfso-relaed ADS daa for llsrave prposes Ke words: Arhmedea opla model odoal lkelhood Prod-lm esmaor Kedall s a Trao daa

2 rodo osder he sao ha a par of falre mes a be lded he sample ol f The varable s sad o be rgh raed b or s lef raed b Mos lerare for aalzg raed daa fos o margal aalss der he assmpo ha ad are qas-depede Tsa 99 However hs odo ma o hold prae For eample he sd of rasfso-relaed ADS he bao me s rgh raed b he lapse me measred from he me of feo o he ed of he sd agakos e al 988 Applg Tsa s es he hpohess of qas-depedee bewee ad was reeed The srprsg assoao ma be arbed o he hage of medal prae dffere hrole perods ad hee sheds some lgh o he poplao dams of ADS To assess he degree of assoao Tsa 99 modfed Kedall s a b odog o a eve ha garaees he hose pars are omparable der rao Tsa s dea was laer eeded o more omplaed rao srres b Mar ad Beesk 5 eel haeb e al 6 proposed a sem-srvval Arhmedea opla A model sable for desrbg he relaoshp for wo varables wh he rao relaoshp The he developed feree proedres o esmae he opla parameer margal fos ad he rao probabl hs arle we adop he same model framework as haeb e al 6 b propose dffere feree proedres The paper s orgazed as follows Seo revews relaed researh developme The proposed mehods for esmag he assoao parameer ad esmao he margal fos are preseed Seo 3 ad 4 respevel Seo 5 oas large sample aalss The proofs are gve Apped Dealed dervaos are provded he aahed ehal repor Seo 6 he

3 proposed proedre s modfed o ao for eeral esorg Adsme for aalzg daases wh small sample szes s also dsssed Nmeral resls ldg smlao sdes ad daa aalss are preseed Seo 7 oldg remarks are gve Seo 8 Prelmar To smpl he aalss s assmed ha boh ad are oos varables ad hee here are o es he daa Assoao Measres ad Models for Tpal Falre Tme Daa aalss of falre me daa robs measres are sall preferred Kedall s a kow as he rak orrelao oeffe s defed as τ E where > } s he oordae daor for he wo pars ad whh are depede replaos of a radom varables oal assoao a be desrbed b he odds rao fo proposed b Oakes 989: Pr > > / Pr > > θ Pr > > / Pr > > / Pr Pr where ad Noe ha he sg of logθ daes he dreo of assoao a me opla models form a lass of bvarae dsrbos whose margals are form o he erval Gees ad MaKa 986 applaos he opla srre s sall mposed o he srvval fos of he falre me varables sh ha Pr > > Pr > Pr > }

4 where v :[] [] ad he parameer s relaed o Kedall s τ wh τ 4 v d dv A sefl sb-lass of he opla faml s he Arhmedea oplas A model wh v beg smplfed as v v} for v [] where :[] [ ] s a geerag fo sasfg / < ad / > Oakes 989 showed ha for a A model deed b he odds rao a be wre as θ θ Pr > > } where θ s a varae fo sasfg θ v v v / v Noe ha whe log ad are depede ad θ v Assoao Measres ad Models for Traed Daa Whe are observable ol f all he aforemeoed desrpve measres are o defable Tsa 99 proposed a modfed verso of Kedall s a gve b τ a E A where A } ad Noe ha b odog o he eve A wo loaos ad o he plae beome omparable der rao se s loaed he defable rego : < } haeb e al 6 modfed he loal odds rao fo as follows: θ Pr > / Pr > Pr > / Pr > / a 3

5 Pr Pr b The erpreao of θ s smlar o / θ b ol he former s defable for raed daa lgh of eqao a haeb e al 6 sggesed o mpose he opla srre o he sem-srvval fo Pr > he rego of The proposed he followg sem-srvval A model: [ F } S }] / 3 where Pr > F ad S are arbrar oos dsrbo ad srvval fo respevel ad s a ormalzg osa sasfg [ F } S dd }] A sem-srvval A model deed b has he proper θ θ } where θ v s defed Eqaos Pr S Pr > ad Pr < do o hold F geeral se he margal fos F ad S ol splae he fo he rego More dealed dsssos for he erpreao of margal fos a be fod haeb e al 6 3 Prevos feree esls for Depede Trao Daa rao daa he sample osss of K } sbe o haeb e al 6 osdered a esmaor of τ E A der a sem-srvval models 3 ad se her esmaor o be eqal o he oparamer esmaor 4

6 θ A} θ } A } < < Ths eqao a be sed as a esmag fo for haeb e al 6 rewre he eqao he -sass form o sd he asmpo behavor < A } θ } Now we geeralze he preedg formla he followg wa } w < A } w 5 θ } where w s a wegh fo ad > / Whe w he above fo redes o he eqao based o he odoal Kedall s a Noe ha w 5 also volves he rao proporo Noe ha he speal ase of lao s model wh > ad θ v w depeds ol o Ths mples addoal esmao proedre for s eeded Se ad are depede esg mehods for esmag he rao probabl are o applable He ad ag 998 se he assme qas-depedee haeb e al 6 proposed her seod esmag proedre based o he modfao of eqao 3: F } S } 6 The adoped he dea of ves ad Wells b frs esmag he mps S } S } ad F } F } based o 6 ad he 5

7 smmg hem p over all he falre mes pror o o oba he esmaors for } ad } B plggg all he margal esmaors o eqao 6 F S a esmag fo for a be obaed 3 Esmag he Assoao Parameer for Sem-srvval opla Models 3 Movao Before we rode he proposed mehods s worh o revew he developme of feree mehods for opla models mposed o he srvval fos Earl work fosed o he lao opla lao 978 based o rgh esored daa hs ladmark paper lao 978 obaed a esmaor for he assoao parameer b mamzg a lkelhood fo whh a be epressed as he prod of odoal probables Ths esmaor was laer re-epressed b lao ad zk 985 as a weghed verso of Oakes oordae esmaor Oakes 98 The ew represeao s relaed o a -sass whh s sefl for dervg asmpo properes Oakes 986 feree of opla models has bee eeded o sem-ompeg rsks daa whh oe varable s a ompeg rsk for he oher b o versa og-rak pe esmag fos have bee proposed b Da e al 997 ad Wag 3 for lao s model ad geeral A models respevel Esmag fos sg he oordae formao for pared observaos have bee proposed b Fe e al 3 Esmao based o odoal kelhood We geeralze he dea of lao 978 o rao daa Defe he se of grd pos as follows: 6

8 7 < ϕ For a po ϕ we defe he mber a-rsk e be a bar varable dag wheher falre ors a Gve r for ϕ ad der model 3 he varable follows a Beroll dsrbo wh } } } Pr r r θ θ ϕ Se } Pr ϕ r ma oa ol lle formao abo we a ol se he odoal probabl o osr he followg lkelhood fo: ϕ θ θ θ } } } The sae parameer a be esmaed oparamerall b / Dffereag log wh respe o we ge he followg esmag fo } } } } ϕ θ θ θ θ & 7 where θ θ / v v & For he ase of lao s model redes o ϕ whh depeds ol o However for he oher members he A faml esmao of reqres he formao of For ma models / log elds he same esmag fo as se ν θ depeds o ol hrogh a sgle parameer For eample Frak opla has he form

9 log θ log / e ad s a fo of sgle parameer γ log Ths mples ha he lkelhood fo a o def ad smlaeosl 3 Esmao based o Two-b-wo Tables Movaed b he deas of Da e al 997 ad Wag 3 we a osr a seres of ables a observed falre pos wh : > N d d N d < N d Here he ell os a be deoed as N d d N d ad N d The odds rao of he above able s relaed o θ defed Gve he margal os he odoal mea of N d der model 3 s d N d N d θ } E N d d N N θ } The reslg log-rak pe esmag fo a be wre as N d N d θ } 8a θ } w w N d d where s a wegh fo Whe he daa have o es we have w N d N d f ad ol f ϕ ad N d d Aordgl w ϕ w θ } 8b θ } 8

10 whh mples ha s a speal ase of w wh he wegh w & θ }/ θ } whh lzes he odoal lkelhood formao The preedg wegh fo a also be obaed as he dervave of he odoal mea of N d wh respe o d dvded b he odoal varae of N d Ths s a gdele for he opmal d wegh fo der he depedee assmpo of all ables Godambe oordae-pe Epresso lao & zk 985 have epressed lao s lkelhood esmaor erms of oordae daors for esored daa We ow esablsh smlar relaoshp for rao daa Some algebra allaos eld he followg de: ϕ w θ } θ } w < [ θ }] A } 9 θ } θ } The proof s gve Apped B whh ldes eeral esorg he aalss Hee he proposed esmag fo s a speal ase of w 5 wh w θ& } θ } θ } θ } The esmag fo proposed b haeb e al 6 orrespods o w ad w θ } θ } The above aalss shows ha he hree dffere feree mehods a be fed Wha reall maers s he hoe of wegh Alhogh some gdeles for hoosg he wegh fo have bee sggesed Fe e al o heoreal sfao s 9

11 provded s reasoable o epe ha whh lzes some lkelhood formao dervao of he wegh ma prode more effe resl ha esmag fos wh ad-ho weghs We wll eame hs oere va smlaos 4 Esmao of Margal Fos ad Trao Probabl Now we propose a ew algorhm for a o esmaor of F S e < < be ordered observed pos of K K ad eplag b / eqao 3 we oba F } S } K Ths s a mor modfao of 6 b a ke o perform he sbseqe algorhm Deoe he esmaors of F ad S as F ad S respevel ad assme ha he are sep fos wh mps ol a observed falre pos der model 3 he kow parameers oss of: F K F } S K S To prode a qe solo eqaos are eeded There are ol eqaos whh perm meros solos Who pror formao s reasoable o add wo bodar osras F ad Togeher wh he eqao here are osras whh wold prode a qe solo Fg a arbrar vale for eqao a be regarded as a esmag fo for F S } Solo for F S } a ol be S obaed for he bodar pos For sae he al osra S mmedael gves he solo F / The proedre s sessvel defed for K

12 Sep f orrespods o a observed vale of se } S S } ad } F S } ad f orrespods o a observed vale of se } F F } ad } S F } Sep Se F } for ma o mee he osra F Jol solvg hs eqao ad gves he esmaors of deoed as Sep 3 edo Sep b seg obaed Sep ad he pdae F } S } s eas o see ha he above algorhm a be wre epl formlas: F } < S } where m ad m Also he eqao Sep a be wre as 3 < he ase of qas-depedee wh log eqaos - 3 rede o he de-bell s esmaors ad he aral esmaor of he rao proporo He ad ag 998 s worh o oe ha he represeao of he de-bell s esmaor

13 as a solo o he mome eqao wh log s ew he lerare ompared wh he prod-lm epresso or approah provdes a more geeral esmag sheme whh allows for depedee bewee he wo varables prple mposg oher bodar osras o F ad S prode a dffere solo o eqaos ad B hoosg F ad he epl formla -3 a be derved Noe ha he proposed S esmaors ad 3 are dffere from he resls of haeb e al 6 However eqao s deal o he proposal of haeb e al 6 5 Asmpo aalss der he reglar odos A-A-V lsed Apped A he esmaors whh ol solve w 5 ad 3 are osse ad asmpoall ormal Noe ha w ldes as a speal ase Weak overgee of he margal esmaors s also esablshed The resls are formall saed he followg heorems Theorem adom veor s osse Theorem adom veor / T overges dsrbo o a bvarae ormal dsrbo wh mea ad ovarae mar A BA T where A E & T ] B E ] ad he defos of [ [ ad & are gve A4 Theorem 3 The bvarae sohas proess / S S F F T overgees weakl o he mea-zero Gassa radom feld G T G G

14 he spae D [ } wh he ovarae fo gve A4 for s < Noe ha haeb e al 6 esablshed smlar resls for her esmaor whh solves w wh w b applg properes of -sass However hs approah ma o be applable whe w volves he plgged- esmaor as or ase Here we ake a dffere approah whh a hadle more geeral wegh fos Spefall asmpo lear represeaos of he proposed esmag fos are obaed B applg he foal dela mehod Va Der Vaar 998 heorem 8 ad properes of empral proesses large-sample properes of he proposed esmaors a be esablshed The skeh of he proof s gve Apped A For he deals please refer o he aahed ehal repor Se he aal dervaos volve omplaed formla we sgges o se he akkfe mehod Spefall o evalae he varae of θ F S he akkfe esmaor s gve b θ θ θ θ where θ s he esmaor gorg h observao ad θ θ / Gve log he odo for qas-depedee he asmpo epresso of A5 of he Apped redes o he d represeao obaed boh Se 993 ad He ad ag 998 Spefall follows ha where / S S S o / F F F d p 3

15 ad > d 6 Eeso ad Modfao 6 Eeso der gh esorg osder a eeded sao ha s sbe o lef rao b ad rgh esorg b aoher varable Assme ha s depede of The sample a be wre as Z δ } sasfg Z where Z δ ad are radom replaos of haeb e al 6 epressed he sem-srvval A model as Pr Z > Z S [ F } S }] / 4 where S Pr > ad s a ormalzg osa sasfg S [ F } S ] dd } 5 The obeve s o esmae he kow parameers F S S Hee we re-parameerze θ } as θ v } where v ad v / S Now we modf he frs esmag fo based o he odoal lkelhood esmao To smplf he preseao we sll se he same oaos b hage her defos as follows The se of grd pos beome: ϕ Z δ Z 4

16 e Z δ Z oseqel he proposed esmag fo s ϕ & θ v } θ v } θ v } θ v } 6 where v / S } ad S s he de-bell s esmaor gve b S / z z δ presee of esorg we a sll epress erms of parwse oordae daors as 9 The dealed epressos are gve Apped B } The proedre for margal esmao a be modfed as follows e < < be ordered observed pos of K Z K Z The esmag fos beome F } S S } K 7 To solve he above eqaos we mpose addoal osras ha he esmaors of F S ad S are sep fos wh mps ol a her observed vales ad ha F S ad The solo a be obaed b performg he followg algorhm sessvel for S Sep f orrespods o a observed vale of se S } S } } F } S S ad S S f orrespods o a observed vale of se 5

17 } F F } } S } F S ad S S ad f orrespods o a observed vale of se F } F } S } S } ad S / } S Sep Se F } o mee he osra F Jol solvg hs eqao ad 6 prodes he esmaors of deoed as Sep 3 edo Sep b seg obaed Sep ad he oba F } S } S Epl formla of he proposed esmaors are gve b z z S } 8 z δ S z S z F } S S 9 < The esmag fo Sep s eqvale o S S < der he speal ase of log whh s he odo of qas-depedee he proposed esmaors S F ad S all rede o he de-bell s esmaors der rgh esorg Apped we derve epl formla 6

18 of he proposed esmag fos for seleed eamples of 6 Modfao for small rsk ses The proposed esmao proedre as well as ha proposed b haeb e al 6 are boh based o he mpl assmpo ha for all However somemes happes ha a emp rsk se ma or espeall he al area Several remedes have bee proposed o hadle hs problem Kle & Moeshberger 3 p Here we adop he dea of a ad g 99 ad propose he followg modfao: S } z δ z z z S z S z a b } where < a < ad b > are arbrar g parameers Modfaos for } ad S are obaed a smlar wa he smlaos o show here F we fod ha akg b ad a / wold prode less based resls 7 Nmeral Aalss 7 Smlao Sdes The ma prposes of he smlao sdes are o hek he vald of he proposed esmaors ad o ompare he performae of or mehod wh s ompeor proposed b haeb e al 6 adom replaos of were geeraed from he lao ad Frak models wh epoeal margal dsrbos sbe o For he lao model he vales of log were hose o be 5 ad 99 ad for he Frak model he vale of log were se o be 38 ad 5746 The former rasformao orrespods o τ 5 ad he laer orrespods o τ 5 The esorg varable also followed a epoeal dsrbo Deoe Pr 7

19 ad Pr For eah seg we repor he bas ad he MSE based o 5 replaos Tables A ad B smmarze he resls for omparg wo esmaors of der he lao model ad Frak model respevel eall ha he proposed mehod solve ad he ompeg esmaor proposed b haeb e al 6 solve w wh w Epl formlas for he lao ad Frak models are derved Apped We see ha boh mehods are appromael based ad he MSE dereases as he sample sze reases ompared wh he esmaor proposed b haeb e al 6 he proposed esmaor has sgfal smaller MSE der lao s model O he oher had he ga effe beomes modes der Frak s model Noe ha he wo approahes prode smlar resls ol der he Frak model absee of eeral esorg fa va eqao we fd ha for Frak s model & θ } θ } os θ } θ } whh eplas wh he meral resls are lose Neverheless whe he esorg proporo reases he proposed esmaor eds o have beer performae Table A ad Table B repor he resls for evalag he proposed seod feree proedre der he lao model ad Frak model respevel The proposed sessve algorhm was arred o ol wh o oba he esmaors of he margal fos ad The performaes of F S were evalaed a seleed pos of wh F 46 8 ad 46 8 Deoe P EN Pr < Z whh measres he esorg proporo he raed S 8

20 sample all he ases F S are farl based s worh o oe ha he esmaed dsrbo/srvval fos have er performae he al area b poorer performae a mddle me po whh behave dfferel from he sal Kapla-Meer esmaor Ths daes ha he al osras F ad S are good appromao of he re vales F S wh sffel large sample szes osdered here 73 Daa aalss We appl he feree proedres o aalze he rasfso-relaed ADS daa dsssed Kalbflesh ad awless 989 e T be he feo me measred form Jaar 978 ad be he bao me from he me of feo o ADS Ol dvdals who developed ADS b he sarg dae Jl 986 old be observed Se he oal sd perod s mohs dvdals wh T were lded he sample whh ossed of 93 sbes Seg T he bao me s rgh raed b Noe ha here was o eeral esorg We aalze he daa der wo dffere model assmpos The resls are smmarzed Table 3 der he lao model boh esmaors of show posve orrelao bewee ad whh mples ha paes who feed earler eded o have loger bao me Boh esmaors reeed he ll hpohess of qas-depedee: H : Ths olso odes wh Tsa s oparamer es 99 The ofdee erval for log based o he proposed lkelhood esmaor s arrower ha ha obaed b he esmaor of haeb e al 6 The level of assoao bewee ad was eve sroger der he Frak model assmpo As he smlaos Table B wh he wo esmaors proded smlar resls Fgre deps he esmaed bao dsrbos der he wo model 9

21 assmpos b applg he proposed rersve algorhm The esmaed rve der he lao model s sgfal lower ha ha der Frak s model mples ha he margal esmaor s also sesve o he model hoe 8 olso Ths arle osders semparamer feree for depede rao daa based o sem-srvval A models proposed b haeb e al 6 parlar we geeralze lao s odoal lkelhood approah o esmag he opla parameer der rao We also geeralze wo-b-wo able approahes of Wag 3 o osr more geeral esmag fos Frhermore we show ha he hree dffere feree mehods a be fed der he same framework The esmaor proposed b haeb e al 6 s osred based o he frs mome odo whh does o oa era formao for hoosg he wegh O he oher had he proposed odoal lkelhood approah provdes heoreal sfao for he mposed wegh ad hee ma prode more effe resls ha oher ad-ho hoes of weghs Aoher poeal advaage of he odoal lkelhood approah s ha a drel hadle he sao whe he dmeso of eeeds oe The proposed sessve algorhm whh solves eqaos ad he wo addoal arfal osras elds smple epl formla ad s eas o mpleme dervaos of he large-sample properes we appl he foal dela mehod whh a hadle more geeral esmag fos ha he -sas approah A seres of smlaos shows ha he proposed odoal lkelhood mehod derease he MSE ompared o he ompeg approah of haeb 6 Neverheless s sll hard o derve a opmal proper of he odoal lkelhood esmaor se s osred based o a workg assmpo of depedee amog he grds wll

22 be a eresg op o frher esablsh some heoreal fodao for hs kd of odoal lkelhood fos der opla models Akowledgemes: Ths work was faall sppored b he seod ahor s researh gra fded b he Naoal See ol of Tawa

23 A- A parameer spae Apped A: eglar odos Θ for s ompa A- Deerms fos v v v / v v v / v v & v ν / v θ θ v θ v / v θ v θ v / v & θ v θ v / w v ad w v w v / v are we oosl dffereable ad boded fo of v A- There ess a fo w } : sh ha sp w w } o / p A-V There ess wo posve mbers < sh ha F > S F ad S > A-V The mar A s o-sglar whose defo s gve laer Noe ha A-V s a odo for he defabl of F S whh has bee roel sed heoreal aalss of rao daa For eample he pper lm plas he same role as he oao T Wag e al 986 A: A Skeh of Asmpo Aalss To smplf he oaos we defe he followg qaes g v / θ v} pr > ad Pr Also le D [ } be a spae ossg of rgh-oos fo T f f wh lef-sde lms where f k : [ a for k The mer s defed as d f g masp < f g k k k } for f g D[ } Smlarl he spae

24 3 } [ D osss of rgh-oos fo s f wh lef-sde lms where s f a [ : eqpped wh he sal sp-orm e Θ be he parameer spae for ad Θ s deoed as he re parameer vale Hereafer epeao smbols represe he odoal epeao gve All he esmag fos ad esmaors hs paper a be appromael epressed as a Hadamard dffereable fo of he empral proess / The foal dela mehod s appled based o he weak overgee resl of / o a Gassa proess W o } [ D wh he ovarae srre gve b: } ov W W for ad The esmag fo w a be epressed as: / Φ Φ o p w A where } [ D }] [ } d d g w > Φ ad Φ h h }] } [ } } } }] } }[ d dh g w d d g w g w > >

25 4 Smlarl a be epressed as / Ψ Ψ o p A where Ψ } } d ad Ψ d } } < d } Boh erms of Φ ad Ψ have zero-meas for a vale of Defe he followg oaos: Ψ Ψ Φ Φ A4 / & ad ] [ E A & From A ad A we have / / o p Ths formla mples ha s a appromae Z-esmaor Va Der Vaar 998 p46 for he rero fo a The osse of follows b hekg he wo odos: The po s he qe zero for ] [ E ad he se of fos } Θ s Glveko-aell Va Der Vaar 999 p46 We also oba / p o A

26 5 Ths he saeme of Theorem holds b leg ] [ T E B For he margal esmaors we derve he followg asmpo lear epresso: } } } } / / p T T o A h h F F S S where T d d H } } ψ ψ T d d H } } ψ ψ ad ad eqal d d } } } } d d } } } } respevel The erms he smmao are d mea-zero sohas proess ad he smmao s a gh proess The oao p o holds forml for [ The we show ha / / p o V V F F S S A5 where } } } T A S S h S V & ad

27 6 } } } T A F F h F V & are mea-zero d sohas proesses ad s smmao s a gh proess e T F F S S V / Also le T G G G be a zero-mea Gassa radom feld he ovarae fo beg spefed as ] [ ] [ V s E V G s G E ] [ ] [ V s E V G s E G ad ] [ ] [ V s E V G s G E for < s Based o he eral lm heorem he fe dmesoal dsrbo of V overges weakl o ha of G ad he ghess proper of V we a prove Theorem 3 Apped B: Eqvalee of Dffere Esmag Fos e } Z B [ ] } } } Z Z Z Z > > δ δ δ δ δ δ be he eve ha he par s orderable ad omparable Mar ad Beesk 5 We am o esablsh he followg de: ϕ θ θ } } v v w < Z v Z Z v Z w B } }] [ } θ θ } Z v θ As a speal ase wh he above de elds eqao 9 The followg proof s for he geeral sao ha perms eeral esorg e

28 θ θ v } ad w w Wrg he egral va he fe sm we oba θ < Z δ w Z N d dz Z θ Z : Z Z < δ w Z Z θ Z : < Z < Z < Z θ Z θ The frs erm a be wre as Z δ w Z θ Z Z θ Z : < Z < Z θ Z δ w Z Z } δ w Z The above eqao follows b og ha he mber of sasfg Z > Z < s Z ad sg he oao ad Z s eas o see ha δ w : < Z < Z < Z δ w Z θ Z θ Z : < Z < Z θ Z Z θ Z B ombg hese erms we have δ δ w Z : < Z < Z θ Z < < B B θ w Z θ Z } } Z θ Z w [ }] Z θ v Z Z θ v Z } Z θ v Z } Apped : Eamples For llsrao we derve epl formlas for he lao ad Frak models 7

29 8 : lao model lao 978 The lao opla s deed b > ad b eqao θ v The sem-srvval lao model follows ha }] [ma / Pr > S F Noe ha he above epresso also aommodaes he ase of < < where < Nelse 999 p9 B eqaos or 5 θ b s erpreao s he reproal of he sal odds rao Hee whe < < we have < θ whh mples posve assoao bewee ad The proposed esmag fo s gve b ϕ ad b solvg a be obaed who esmag or The seod esmag fo redes o he epl formla < S S Plggg he above eqao we oba ĉ The rersve algorhm elds he followg margal esmaors: δ z z S z z S z S < S S F

30 9 : Frak model Frak 987 The Frak opla s deed b } / log > wh / log v v v θ The sem-srvval Frak s model a be wre as ] / [ log / Pr < > S F follows ha } v θ θ / log } v v osder he rasformao log γ The lkelhood esmag fo a be epressed erms of γ ad he proposed esmag fo of γ s gve b ϕ γ γ γ γ γ } } v v e v w where / v v e e v w γ γ γ γ e γ be he solo o γ The assoao parameer a be esmaed b < S S e e e } } / } / / γ γ γ ad hee log / γ Epl formla for he margal esmaors are gve b } / } } / log z z S z z S z S δ < S S F } }/ } / / log eferees haeb ves -P & Abdos B 6 Esmag srvval der a depede rao Bomerka

31 lao D G 978 A model for assoao bvarae lfe ables ad s applao epdemologal sdes of famlal ede hro dsease dee Bomerka lao D G & zk J 985 Mlvarae geeralzaos of he proporoal hazards model wh dssso J Sas So A Da Bra J & efkopolo M 997 Adapao of bvarae fral models for predo wh applao o bologal markers as progos daors Bomerka Fe J P Jag H & happell O sem-ompeg rsks daa Bomerka Gees 987 Frak s faml of bvarae dsrbos Bomerka Gees Ghod K & ves -P 995 A sem-paramer esmao proedre for depedee parameers mlvarae famles of dsrbos Bomerka Gees & Maka J 986 The o of oplas: Bvarae dsrbos wh form margals The Amera Sasa Godambe V P 99 Esmag Fos Oford: laredo Press He S ad ag G 998 Esmao of he rao probabl he radom rao model Aals of Sass 6-7 Kalbflesh J D & awless J F 989 feree based o rerospeve aserame: a aalss of he daa o rasfso-relaed ADS J Am Sas Asso Kle J P & Moeshberger M 3 Srvval Aalss: Tehqes for esored ad Traed Daa New ork: Sprger agakos S W Barra M & De Grola V 998 No-paramer aalss of 3

32 raed srvval daa wh applao o ADS Bomerka a T & g Z 99 Esmag a dsrbo fo wh raed ad esored daa Aals of Sass akhal ves -P ad Abdos B 8 Esmag Srvval ad Assoao a Semompeg sks Model Bomers de-bell D 97 A mehod of allowg for kow observaoal seleo small samples appled o 3 qasars Mo Na Asr So agakos S W Barra M & De Grola V 998 No-paramer aalss of raed srvval daa wh applao o ADS Bomerka Mar E & Beesk A 5 Tesg qas-depedee of falre ad rao va odoal Kedall s Ta J Am Sas Asso Nelse B 999 A rodo o oplas New ork: Sprger-Verlag Oakes D 98 A model for assoao bvarae srvval daa J Sas So B Oakes D 986 Sem-paramer feree a model for assoao bvarae srvval daa Bomerka Oakes D 989 Bvarae srvval models ded b frales J Am Sas Asso ves -P & Wells M T A margale approah o he opla-graph esmaor for he srvval fo der depede esorg J of Ml Aal Shh J H & os T A 995 feree o he assoao parameer opla models for bvarae srvval daa Bomers Se W 993 Almos sre represeao of he prod-lm esmaor for raed daa Aals of Sass

33 Tsa W - 99 Tesg he assoao of depedee of rao me ad falre me Bomerka Va Der Vaar A W 998 Asmpo sass ambrdge Seres Sass ad Probabls Mahemas ambrdge: ambrdge vers Press Wag M Jewell N P & Tsa W 986 Asmpo properes of he prod-lm esmae ad rgh esored daa A Sas Wag W & Dg A A O assessg he assoao for bvarae rre sas daa Bomerka Wag W 3 Esmag he assoao parameer for opla models der depede esorg J Sas So B Woodroofe M 985 Esmag a dsrbo fo wh raed daa A Sas Zheg M & Kle J 995 Esmaes of margal srvval for depede ompeg rsks based o a assmed opla Bomerka

34 Table A: omparso of Two Esmaors for he Assoao Parameer der he lao Model - log τ 5 5 Proposed haeb Proposed haeb Eah ell oas he bas 3 ad MSE parehess of he orrespodg esmaor based o 5 replaos 33

35 Table B: omparso of Two Esmaors for he Assoao Parameer der he Frak Model log 5 5 τ Proposed haeb Proposed haeb Eah ell oas he bas 3 ad MSE parehess of he orrespodg esmaor based o 5 replaos 34

36 Table A: The proposed esmaors of margal fos ad rao proporo der lao Model τ 5 parameer Tre 5 P EN 5 EN P 4 5 EN P 5 P EN 4 / 86/ F F F F S S S S Eah ell oas he bas 3 ad MSE parehess based o he rersve esmaor sg he lkelhood mehod for he assoao parameer The esorg proporo s deoed b Pr < Z P EN 35

37 Table A: The proposed esmaors of margal fos ad rao proporo der Frak Model τ 5 parameer Tre 5 P EN 5 EN P 39 5 EN P 5 P EN 39 / 88/ F F F F S S S S Eah ell oas he bas 3 ad MSE parehess based o he rersve esmaor sg he lkelhood mehod for he assoao parameer The esorg proporo s deoed b Pr < Z P EN 36

38 Table 3 Aalss of he Trasfso-elaed ADS Daa Proposed haeb Assmpo: lao opla Esmaes 95% akkfe erval Esmaes 95% akkfe erval log 3 τ τ Wald s h-sqare for H : log p-vale< p-vale 3 Assmpo: Frak opla Esmaes 95% akkfe erval Esmaes 95% akkfe erval log 375 τ τ Wald s h-sqare for H : log 4696 p-vale< 4495 p-vale< 37

39 F der Frak der lao Fg The mlave dsrbo fos of he bao me of ADS der wo opla models 38