Asymptotic inference for MLEs under the special exponential family with double-truncation. Takeshi Emura( 江村剛志 )

Transcription

1 Asmtotc erece or MLEs uder the secal exoetal aml wth double-trucato Takesh Emura 江村剛志 Graduate Isttute o Statstcs Natoal Cetral Uverst Jot work wth Ya-Hsua Hu Yoshhko Koo /5/5 Ya-Hsua Hu 6//4

2 Outle Presetato s based o our aers: [] Emura T Hu YH & Koo Y 5+ Asmtotc erece or maxmum lkelhood estmators or a cubc exoetal aml uder radom double-trucato major revso submtted to Statstcal Paers [] Hu YH & Emura T 5 Maxmum lkelhood estmato or a secal exoetal aml uder radom doubletrucato Comutatoal Statstcs. DOI.7/s z [] Hu YH 4. Maxmum lkelhood estmato or doubletrucato data uder a secal exoetal aml. NCU Master Thess

3 Chldhood cacer data Morera ad de Uña-Álvarez Y U V : Age at cacer das : Age at recrutmet start das : Age at recrutmet ed moths U 85 V U Cacer Brth da 999// Recrutmet Start // Recrutmet Ed Y 5 ear 85 das ollow-u 6/4

4 Double-trucato: U Y V wth U V Y I U Y V observed; otherwse othg s avalable! Observato: subject to u v Target o Estmato: S P Y d P Y d

5 Noarametrc aroaches to double-trucato Ero ad Petrosa 999 JASA - Noarametrc maxmum lkelhood estmator NPMLE. She AISM - Uorm cosstec ad weak covergece o NPMLE Morera ad Uña-Á lvarez J o Noar - Bootstra codece terval Morera ad Kelegom CSDA -A kerel dest estmato She J. Noar Aust et al. 4 LIDA - Ideedece test U V Y Emura et al. 5 LIDA - Exlct ormula o asmtotc varace o NPMLE 5

6 Parametrc aroaches to double-trucato Ero ad Petrosa 999 JASA -Maxmum lkelhood estmator MLE uder the seres exoetal aml SEF. Hu ad Emura 5 Comutatoal Statstcs -Newto Rahso algorthm to obta the MLE uder the SEF Motvato: Asmtotc theor o the MLE uder double-trucato has ot bee studed Asmtotc theor does ot ollows rom the usual cetral lmt theorems due to double-trucato Wh? Data are ot..d 6

7 6//4 Secal Exoetal aml SEF troduced b Ero ad Petrosa 999 JASA Letme ollows a cotuous dstrbuto wth a dest. } ex{ T t : the suort o T k t k T k ] } ex{ log[ T d t 7 Y Y I ocus o k

8 Secal Exoetal aml k ex[ ] ] Parameter sace :. } { Uer boud or the suort max maxmum observed 6//4 letme

9 Secal Exoetal aml k ex[ ] [ Parameter sace :. } { Lower boud or the suort m 9 mmum observed letme 6//4

10 Secal exoetal aml SEF: Summar ex[ k k ] Dstrbuto arameter SEF Locato-exoetal tal dstrbuto arameter SEF Locato-Scale exoetal dstrbuto arameter SEF Normal dstrbuto Skew arameter SEF U-shaed dstrbuto arameter cubc SEF Postvel skewed ormal dstrbuto arameter cubc SEF Negatvel skewed ormal dstrbuto Resemble to lexble skew ormal class Re: Chag ad Geto 7 Comm. Stat. T&M ad ma other skew ormal dstrbutos 6//4

11 Lkelhood uder double-trucato Trucato terval: R [ u v ] u No lkelhood to be observed outsde the terval! Trubull 976 JRSSB; Ero ad Petrosa 999 JASA v Trucated dest codtoal dest gve R { R F where F v u d 6//4 }

12 6//4 Lkelhood uder double-trucato Log-lkelhood: Uder the cubc SEF k=: F } log { log log. ex log ex log v v u d d Postveskew ;

13 6//4 Maxmum lkelhood estmato- Cubc SEF k Newto-Rahso Algorthm Ste : Choose the tal value. T Ste : Reeat utl covergece The s the MLE... crtero : Stog

14 Maxmum lkelhood estmato- Cubc SEF k Radomzed Newto-Rahso Algorthm Hu ad Emura 5 Comutatoal Statstcs: Ste 4: I the Newto Rahso dverges.e. D D or D or Dameters D D D Retur to Ste : b relacg T wth T u u where u U d ad u U d ~ d We chose d 6 ad d ~ d 6//4

15 6//4 Estmator o dest Estmator o survval ucto We alwas eed SE ad codece tervals or ad. } ex{ t t t t t 5 } ex{ dt t t t S Maxmum lkelhood estmato- Cubc SEF k MLE : j j ex log where d t S

16 Asmtotc aalss o the MLE MLE : j... k j Exstece o the MLE Cosstec Asmtotc ormalt SE ad codece terval have ot bee studed Our goal: Asmtotc aalss based o deedet but ot detcall dstrbuted...d data. 6 6//4

17 6//4 Lkelhood uder double-trucato Lkelhood: Itervals are heterogeeous betwee samles Score ucto: ot the sum o..d. terms 7 } { v u F v u v u } log { log log N F d CLT?

18 Asmtotc theores o the MLE uder...d. Bradle ad Gart 96 Bometrka: -Semal work or the MLE uder...d. -Proos are ot rgorous the lterature 94 s durg whch the robablt theor was ot establshed Hoadle 97 A. Math. Stat - Regulart codto arl techcal ad less tutve - Examle o letme model wth xed cesored ots Phlou ad Roussas 975 AISM - Cosstec o MLE s assumed at the begg. Cocluso: Above exstg theorems are ot relable ad helul to our case. Yet the dea o Bradle ad Gart 96 s aealg 6//4 8

19 Our strateg o establshg asmtotcs Regulart codtos: Follow the stles o Bradle ad Gart 96 Bometrka: but ot ollow ther roos Tools: Textbooks o mathematcal statstcs: - Strog Law o Large Number or...d. Shao - Ldeberg-Feller Cetral Lmt Theorem va der Vaart 998 Proos: Mod the roo o Lehma ad Casella 998 or..d. case - Hadlg mult-arameter cases - Rgorous + Clear - Smultaeousl establsh: Exstece + Cosstec + Asmtotc ormalt 6//4

20 Strog Law o Large Number SLLN or...d. Lemma Let or Y Y be deedet radom varables wth E [ Y ]. I there s a costat [ ] s.t. lm E[ Y ] P the YE[ Y ]. Ldeberg-Feller Cetral Lmt Theorem CLT Lemma Let D be deedet k-dmesoal radom vectors s.t. D E[ D E[ D ] { D E[ D ] } ] Ldeberg Codto or ever ad Cov D d The D E D N as. k 6//4

21 Characterzato o the MLE Dee to be a soluto to the score equatos T k j j k Assumto A The arameter sace s oe ad cotas the true k T. Parameter sace s atural.e. ex{ T t } d. Lemma Characterzato o MLE: Uder Assumto A the soluto exsts the t s the MLE T k.e. or a. 6//4

22 Fsher ormato matrx or...d. data Assumto B There exst a k k ostve dete matrx I { I js } j s... k s.t. I js / I js j s {... k } as. where the Fsher ormato o the th samle s I js E log Y log Y j s. j s k 6//4

23 Boudeess codtos: Assumto C For j s l {... k } there exst M s.t. jsl log j s l M jsl m E { M Y } ad m E { M Y }. jsl jsl jsl jsl Assumto D For j s {... k } there exst W js s.t. log j s W js w E { W Y } ad w E { W Y }. js js js js Assumto E For j {... k } there exst A s.t. j log j A j wth su A j. Assumto E: Smlar to Bradle ad Gart 96 Bometrka to regulate...d. samles 6//4

24 6//4 4 Ma result: Asmtotc theor Theorem : I Assumtos A-E hold the a Exstece ad cosstec: There exsts a soluto wth robablt tedg to oe s.t. P as. b Asmtotc ormalt: I N k d as. Vald aroxmato to the Fsher ormato. log log log js s j s j s j js js I Y E Y Y E I I

25 6//4 5 Asmtotc erece or dest Stadard Error SE T } { Where / / e e e e k k ad where d e j j } ex{ T t }... { k j. % codece terval or s }] { } { [ / / SE Z SE Z.

26 6//4 6 Asmtotc erece or survval ucto Stadard Error S S S SE T } { Where dt t e e t e e t S t k k / /. % codece terval or S s }] { } { [ / / S SE Z S S SE Z S.

27 v u Eas-to-check sucet codtos Lemma 4 Assumtos C D ad E hold uder the ollowg two codtos: Assumto D The arameter sace s bouded. Assumto G The lower suort o let-trucato u u The uer suort o rght-trucato v su v. su There exst costats u v such that [ u v] [ u v ] [ u vsu]. Iterretato o Assumto G Boudedess o trucato terval Sucet ollow-u: Iterval caot be too short should be loger tha v u 7 6//4

28 v u Smulato results the cubc SEF based o reettos uder the cluso robablt P U Y V. 5. E SD E SE ] 95%Cov 8 [ E SD E SE ] 95%Cov [ E SD E SE ] 95%Cov [ //4

29 v u Smulato results the cubc SEF based o reettos uder the cluso robablt P U Y V. 5. E{ S t } SD{ S t } E[ SE { S t }] 95%Cov S S E{ t } SD{ t } E[ SE{ t } ] 95%Cov //4

30 Data aalss o chldhood cacer data Morera ad de Uña-Álvarez Y U V : Age at cacer das Estmato : Age at recrutmet start das : Age at recrutmet ed moths U 85 V U 999// Recrutmet Start // Recrutmet Ed Y 5 ear 85 das ollow-u 6/4

31 Outle Itroducto Methodolog Theor Data aalss Cocluso Reerece Frst model selecto Kolmogorov-Smrov dstace D max{ S S } NPMLE SNPMLE P Y : Model-ree survval ucto where s the NPMLE Ero ad Petrosa 999 S P Y : Model-based survval ucto AIC Akake Iormato Crtero k AIC log L k : the umber o ukow arameters : maxmzed value o lkelhood ucto log L

32 KS-dstace betwee MLE ad NPMLE Best Model = Smallest KL dstace = Cubc SEF wth 6//4

33 Data aalss -- Model selecto -- The maxmum lkelhood erece or the chldhood cacer data. Model log L AIC K-S statstc 5 a ar. SEF b ar. SEF c ar. SEF d Cubc SEF Model a = The oe-arameter SEF Model b = The oe-arameter SEF Model c = The two-arameter SEF Best model Model d = The cubc SEF log L = The maxmzed log-lkelhood AIC = Akake ormato crtero deed as AIC log L k K-S statstc = The Kolmogorov-Smrov dstace betwee the MLE ad the NPMLE 6//4

34 6//4 Data aalss uder the cubc SEF best model 4 d d t S t t ] ex[ mmum observed letme m

35 Asmtotc erece uder the cubc SEF ex[ ] : Estmate o the dest : 95% Codece terval Hgh rsk o develog cacer earl ages : Same dg oud Emura et al. 5 LIDA 5 6//4

36 Pck u reerees commets : Statstcal Paers Q: Assumto G hold or real data examle? Aswer Follow-u legth: Fxed at 5 ears 85 das. But Assumto G requres d d 7 das So Assumto G does ot hold. But t s eas to be checked b user Other target quattes uder double-trucato S t w t S Y t w Y t Predctve survval Kle & Moeschberger wth let-trucato ol Mea / meda resdual le m u E Y u Y u Ch et al. 4 Com.Stat-Smulatos wth let-trucato ol m u v E Y v u Y v 6 6//4 wth double-trucato Sakara & Suoj 4 Stat Paers

37 Outle Itroducto Methodolog Theor Data aalss Cocluso Reerece Thak ou or our lsteg 7