Approximation of the Likelihood Ratio Statistics in Competing Risks Model Under Informative Random Censorship From Both Sides

Transcription

1 Approimatio of the Lielihood Ratio Statistics i Competig Riss Model Uder Iformative Radom Cesorship From Both Sides Abdrahim A. Abdshrov Natioal Uiversity of Uzbeista Departmet of Theory Probability ad Mathematical Statistics 0078 Tashet Uzbeista a_abdshrov@rambler.r Nargiza S. Nrmhamedova Natioal Uiversity of Uzbeista Departmet of Theory Probability ad Mathematical Statistics 0078 Tashet Uzbeista raslova_argiza@mail.r Abstract It is clear that the lielihood ratio statistics plays a importat role i theories of asymptotical estimatio ad hypothesis testig. The aim of the paper is to ivestigate the asymptotic properties of lielihood ratio statistics i competig riss model with iformative radom cesorship from both sides. We prove the approimatio versio of the locally asymptotically ormality of the lielihood ratio statistics. The reslts have asymptotic represetatio of the lielihood ratio statistics sig the strog approimatio method where local asymptotic ormality is obtaied as a coseqece. Keywords: Lielihood ratio statistics Competig riss model Locally asymptotically ormality Radom cesorig.. Itrodctio The most importat property of statistical models is the locally asymptotically ormality (LAN) of lielihood ratio statistics (LRS) of a reglar statistical eperimet. The essece of the LAN is that the LRS of model ca be approimated by fctios of the from ep where - asymptotically (at ) ormal with the parameters (0) radom variables (r.v.). LAN i the case of idepedet ad idetically distribted (i.i.d.) observatios stdied by A.Wald L.Le Kam ad J. Hae (for details see [6-]). I [3-5] established reslts of strog approimatio for LRS by stochastic itegrals i competig riss model (CRM) for radom cesorig of observatios o the right ad from both sides. I this paper we cosider similar problems i the CRM whe radom cesorig from both sides is iformative.. Iformative model ad its characterizatio Followig [3-5] we cosider the CRM. Let X - r.v. with vales i a measrable space () ( ) ( XB ) where X R B ( X ). Cosider the disoit evets { A... A } (or at least Pa..stat.oper.res. Vol.XII No. 06 pp55-64

2 Abdrahim A. Abdshrov Nargiza S. Nrmhamedova P A i i ) sch that P A. I CRM or iterest is i ( i) ( ) ( A ) 0 cocetrated o oit properties of r.v. X ad evets A i. Let ( i) ( i) I A i - idicators of these evets ad oit distribtio of the vector { ( ) } () ( ) X depeds o ow parameter : (... ) () ( ) () () ( ) ( ) Q( y... y ) P( X y... y ) ( ) where R y i {0} i ad a ope set i parameter. R for simplicity is scalar Let H( ; ) P ( X ) ad X ad H ( ; ) - margial distribtios of r.v. P ( X ) () ( ) ( X ) i respectively. Sice... the () ( ) ( ; )... H H ( ; ) H( ; ) - for all ( ; ) R. Let sbdistribtios H are absoltely cotios ad have desities h. The there is a desity h of d.f. H ad for all () ( ; ) R : h( ; ) h ( ; )... ( h ) ( ; ). Assme that the r.v. X sbect to radom cesorig from both sides by pair of r.v.-s ( LY ) with absoltely cotios d.f.-s K( ; ) P ( L ) ad G( y; ) P ( Y y). It also assmes that the r.v.-s { X L Y } - are idepedet ad cesorig is iformative i.e. d.f.-s K ad G epressed i terms of d.f. H by the followig formlas for all ( ; ) R : G( ; ) H ( ; ) K( ; ) H( ; ) where ad are positive ad ow isace parameters idepedet of. Let ad g are desity fctios correspodig to the d.f. K ad G respectively. The accordig to () H H ( ; ) ( ; ) ( ) ( ; ) h( ; ) g( ; ) H( ; ) h( ; ). () Let H if : H( ; ) 0 ad TH sp : H( ; ). The from () H K L ad a desities () are positive o the set [ ; T ]. H H () TH TK TL () ( ) Let ( X L Y A... A ) - a seqece of idepedet replicas of the aggregate () ( ) Y. O - th stage of the eperimet we observe the sample of size : ( X L A... A ) ( ) ( ) (0) () ( ) (... ) where ( ;... ) L ( X Y ) ma( L mi( X Y )) ID ( ) l ( l) ( l) (0) ( ) ad evets D : X ( ) Y ( ) L ( ) i i D A : L ( ) X ( ) Y ( ) i... ( ) ( ) D : L ( ) Y ( ) X ( ). We 56 Pa..stat.oper.res. Vol.XII No. 06 pp55-64

3 Approimatio of the Lielihood Ratio Statistics i Competig Riss Model Uder Iformative Radom. itrodce d.f.-s M ( ; ) P X Y ( G( ; )) ( H( ; )) ad N( ; ) P ( l) ( l) K( ; ) M( ; ) as well as sbdistribtios T ( ; ) P ; l easy to see that ( T ) ( ; ) M ( ; ) dk( ; ) It is T (0) ( ; ) K( ; )( H( ; )) dg( ; ) (3) ( i) ( i) T ( ; ) K( ; )( G( ; )) dh ( ; ) i. The T ( ; ) K( ; )( G( ; )) dh ( ; ) T i ( ; ) ( ) (0) () ( T ( ; ) T ( ; ) T ( ; )... T ) ( ; ) N( ; ). Let ad accordig to () by direct calclatio of sbdistribtios (3) we have ad. The M ( T ) ( ; ) M ( ; ) d M ( ; ) ( ; ) N( ; ) (4) (0) T ( ; ) M ( ; ) H( ; ) d H( ; ) M ( ; ) d H( ; ) M ( ; ) dm ( ; ) T( ; ) M ( ; ) d H( ; ) N ( ; ) (5) M ( ; ) dm ( ; ) N( ; ). (6) Iformative model der simple radom cesorig from both sides withot competig riss was first itrodced i []. Followig [] we prove a theorem characterizig the ( ) (0) model () by idepedece of r.v.-s where () ( ).... ad vectors Theorem.. Model () holds if ad oly if the r.v. idepedet for each. ad vector ( ) (0) are Proof of the Theorem.. Sppose that the formlas () are hold. The we have the represetatio (4) - (6) for sbdistribtios. Tedig to the limit der i these formlas i particlar for all we have P ( ) (0) P ( ) P. (7) Pa..stat.oper.res. Vol.XII No. 06 pp

4 Abdrahim A. Abdshrov Nargiza S. Nrmhamedova Itrodce the evets A B ad B B B ( m) 0 where Bm m 0 ad. The from (4)-(7) it follows the idepedece of evets A ad Bm m 0. Similarly we ca show idepedece of other combiatios of these ( ) (0) evets ad their reectios. This shows that the r.v. ad vector are ( ) (0) idepedet. Coversely let r.v. ad vector are idepedet. I particlar from idepedece of K ( ; ) ep ( ) P M K ( ; ) ( ; ). ad ( ) for all K we have ( ) dt ( ; ) ( ) dn( ; ) ep P M ( ; ) K( ; ) N ( ; ) ( ) ( ) From here the secod represetatio i () follows der P P O the other had sice idepedece of ad (0) for all G. dt (0) ( ; ) G ( ; ) ep K( ; ) H( ; ) (0) dn( ; ) ep P K( ; ) N( ; ) (8) also from idepedece of ad for all we have dt ( ; ) H( ; ) ep dn( ; ) ep P. K( ; ) N( ; ) K( ; ) N( ; ) (9) Now from (8) ad (9) for all ( ; ) R G H log ( ; ) log ( ; ) (0) (0) where P P which shows the validity of first formla i (). Theorem. is proved. ( ) ( ) p Let p P ad p P. The p ( ) p ad therefore these parameters are estimated by the statistics p ( ) ad ( ) p () p ( ) ( ) where p p i. H ( ) 3. Approimatio of the LRS 58 Pa..stat.oper.res. Vol.XII No. 06 pp55-64

5 Approimatio of the Lielihood Ratio Statistics i Competig Riss Model Uder Iformative Radom. Let ( ) ( ) ( ) Q ( Y U ) - deote the seqece of statistical eperimets geerated by observatios ( ) where ( ) ( ) ( ) Y { X {0} } ( ) ( ) Y U. The family of measres ( ) respect to measre measres cocetrated at a poit by ( ) ( ) dq p ( ; ) d ( ) ( ) ( ) U ( Y ) ( ) Q - distribtio o ( ) { Q } is absoltely cotios with ( )... where d d ( )... ( ) ( l ) - cotig ( ) ( ) ( ) ( ) () l m m m y y m y {0} l 0... ; m ad its desity is give ( ) M ( ; ) ( ; ) K( ; ) G( ; ) h ( ; ). i m y m (0) K( ; ) H( ; ) g( ; ) Now by sig () ad () after simple algebra the desity () ca be represeted as follows: ( ) p ( ; ) ( ; ) t( ; ) t ( ; ) (3) i where ( ; ) ( ) (0) ( ) - idepedet of i () t( ; ) t ( ; ) - desity of sbdistribtio T ( ; ) ad t ( ; ) h ( ; ) K ( ; ) G ( ; ) T ( ; ) i. Note that accordig to the represetatio (3) statistics ; is a sfficiet statistic for this iformative model. From (3) oe ca epect that the LAN property for this model depeds o the properties of the desity t i.... Becase ( i) ( i) H( ; ) H( ; ) the have ( i) ( i) t ( ; ) h ( ; ) ad t ( ; ) h ( ; ) t( ; ) h( ; ) for all i ( ; ) R. We eed some reglarity coditios: (C) Spports N ( i ) { : h ( ; ) 0} i are idepedet o parameter ad i N ( i ) h ; h (C) For ay ad N h ( i ): h ( ; ) h ( ; ) i... ; ( i) ( i) (C3) There are fiite for all derivatives l h ( ; ) / l l ; i ad () l i ( ; ) / l ;... h d l i ; (C4) (C5) Fctios log t ( ; ) log t( ; ) Fisher iformatio i are of boded variatios; Pa..stat.oper.res. Vol.XII No. 06 pp

6 Abdrahim A. Abdshrov Nargiza S. Nrmhamedova J () i log t ( ; ) ( ) dt ( ; ) i is fiite ad positive at the poit 0. ( ) (0) d T T log t ( ; ) ( ; ) ( ; ) We defie the empirical estimates of the distribtios N ( ; ) ad T ( ; ) : N ( ) I T ( ) T ( ) T ( ) ( ) (0) ( m) ( m) T ( ) I T ( ) T ( ). i ( m T ) ( ; ) m 0... m 0... ( m) ( m) We deote N( ; 0) N( ) T ( ; 0) T ( ) m 0... ad T( ; 0) T( ) where 0 is tre vale of. Let 0 where R. We itrodce the LRS ( i ) () ( ) i p ( ; ) ( ; ) t t ( ; ) L ( ) p ( ; 0) i t ( ; 0) t ( ; 0) ad its logarithm log t ( ; ) ( ; ) L log i t 0 t ( ; ) log t ( ; 0 ) We have t ( ; ) log dt ( ) i t ( ; 0) t ( ; ). (4) ( ) (0) log d T ( ) T ( ) t ( ; 0) Theorem 3.. Let the reglarity coditios (C) - (C5) are hold. The for each the LRS we have represetatio where L ep{ W J ( ) R ( )} 0 R for W i log t ( ; 0) d W ( T ( ); ) / ( i) i log t ( ; 0) / ( ) (0) d W ( T ( ); ) W0( T ( ); ) ad R ( ) 0 at i are two 0 parametrical Wieer processes o [0] (0 ) ad the compoets of the vector ( W W0 W... W ) are idepedet. ( ) Q -probability. Here W m ( y; ) m Pa..stat.oper.res. Vol.XII No. 06 pp55-64

7 Approimatio of the Lielihood Ratio Statistics i Competig Riss Model Uder Iformative Radom. Remar 3.. I view of the properties of processes W 0... m m the r.v. W is the sm of idepedet stochastic itegrals of Ito ad for each : D ( ) / (0 J ( 0 0)) L W Q N. Hece the theorem 3. oe ca writte as follows: for each R / L J ( 0) J ( 0) R( ) (5) D where N (0). To prove Theorem 3. we eed i some ailiary reslts. Defie the empirical processes of R : ( ) ( N ( ) N( )) i ( i) ( i) i( ) ( T ( ) T ( )) We formlate a theorem which is a geeralized aaloge of Komlos Maor ad 3 Tsady s type reslt. Let v ma{ v v... v } - maimm-orm of a - v- v 0 3 vector v ( v- v- v0... ) v v R. From [.9] we give the followig statemet of Bre Csörgő ad Horvath. Theorem A []. O the probability space ( AP) oe ca costrct +3 twoparametrical processes K( ; ) K ( ; ) K ( ; ) K ( ; )... ( ; ) 0 K sch that for the 3 vector ( v) ( ( v ) ( v ) 0( v0 ) ( v )... ( v )) v ( v- v- v0... ) v v R we have approimatio 3 vr 3 a.s. / / sp ( v) ( v; ) O( (log ) ) (6) where ( v; ) ( K( v ; ) K ( v ; ) K0( v0; ) K( v; )... K( v; )) is +3- dimesioal Gassia / process havig the same covariace strctre as a vector () v i.e. E( v; ) 0 ad for ay i 0 i : EK( ; ) K( y; m) ( m){ N( ) N( y) - N( ) N( y)} ( i) ( i) ( i) ( i) EK ( ; ) K ( y; m) ( m){ T ( ) T ( y) - T ( ) T ( y)} i i EK K y m m T T y ( i) ( ) i( ; ) ( ; ) ( ){- ( ) ( )} ( ) ( ) ( ) EKi ( ; ) K( y; m) ( m){ T i ( ) T i ( y) - T i ( ) N( y)}. (7) Lemma 3.. Let the reglarity coditio (C)-(C3) are hold. The there eist fiite derivatives l ( ; ) / l t l ; i for all ( ; ) R ad l t ( ; ) d l ; i... ; (8) l Pa..stat.oper.res. Vol.XII No. 06 pp

8 Abdrahim A. Abdshrov Nargiza S. Nrmhamedova i log t ( ; ) log t( ; ) E E 0. (9) Proof of the Lemma 3.. With regard to the epressio of sbdesity t throgh h i it is easy to see that the reglarity coditio (C) - (C3) are hold for them ad i particlar (8) also holds. Moreover taig ito accot the idepedece of r.v. ( ) (0) ad (Theorem.) we have ( ) log p ( ; ) log t ( ; ) E dt ( ; ) i log t ( ; ) ( p) dt( ; ) t ( ; ) d i ( p) t( ; ) d ( p) T( ; ) ( p) ( ) 0 as ad assmed to be idepedet of. Lemma 3. is proved. Lemma 3.. Uder the coditios of Theorem 3. whe for each R we have L log t ( ; 0) dt ( ) i log t ( ; ) ( ) ( ) 0 ( ) (0) d T T J ( 0) op (). (0) Proof of the Lemma 3.. We epad the itegrads i LRS L ( ) i Taylor series o / t ( ; ) t ( ; ) powers of i ( ) ad ( ). The we obtai respectively t ( ; 0) t ( ; 0) t ( ; ) i t ( ; 0) log log( ( )) / 3 i ( ) i ( ) i ( ) i ( ) () t ( ; ) log log( 3 ( )) ( ) ( ) ( ) ( ) () t ( ; 0) where i ( ) i ad ( ). I other had fctios i ( ) ( ) epad i the eighborhood of 0. We have ( i) ( i) log t ( ; 0) log t ( ; 0) ( i) t ( ; 0) i ( ) ( i) o 8 t ( ; 0) i... ad 6 Pa..stat.oper.res. Vol.XII No. 06 pp55-64

9 Approimatio of the Lielihood Ratio Statistics i Competig Riss Model Uder Iformative Radom. log t( ; 0) log t( ; 0) ( ) 8 t ( ; 0) o. t( ; 0) It is easy to verify that for all i log t ( ; 0) i ( ) o 4 log t ( ; 0) ( ) o 4 i ( ) o 3 ( ) o 3 Ths whe log t ( ; 0) dt ( ) ( i) ( ) (0) i ( ) dt ( ) ( ) d T ( ) T ( ) i log t ( ; ) ( ) ( ) R R o p () 4 0 ( ) (0) d T T where i (3) log t ( ; 0) R dt ( ) i log t ( ; ) 0 ( ) (0) d T ( ) T ( ) R t ( ; ) dt ( ) ( ; ) ( i) ( i) 0 ( i) i t 0 d T ( ) T ( ). ( ) (0) t ( ; 0) t ( ; 0) Note that E R 0 J ( 0) ad i view of ( ) (0) ( i) T ( ; ) T ( ; ) T ( ; ) E R 0 0. Coseqetly accordig to the law of large mbers for R J ( ) o () R o (). 0 p p Similarly for we have i (4) ad ( ) dt ( ) ( ) d T ( ) T ( ) J ( 0) op () (5) 4 ( ) (0) ( ) 3 ( ) (0) i ( ) ( ) d T ( ) T ( X ) i ( ) i ( ) dt ( ) op(). 3 () (6) i Now ()-(6) implies (0). Lemma 3. is proved. Pa..stat.oper.res. Vol.XII No. 06 pp

10 Abdrahim A. Abdshrov Nargiza S. Nrmhamedova Proof of the Theorem 3. is ses Lemmas 3. ad 3. is flly codcted throgh the proof of Theorem i [4] sig also Theorem A ad therefore details are omitted. Coclsio Asymptotic theory of estimatio ad hypothesis testig is etirely based o the asymptotic properties of the LRS. The most importat property of LRS is LAN which eables the developmet of the asymptotic theory of maimm lielihood ad Bayesia type estimators ad cotigity of probability measres. I this paper proved the LAN property of LRS i CRM der radom cesorig from both sides. These reslts ca be sed to fid the limitig distribtios of maimm lielihood ad the Bayesia type estimates i cosidered model. Refereces. Abdshrov A.A. The model of radom cesorship from both sides ad the criteria of idepedece for her // Doclady Acad. Na. Uzbeista p.8-9. (I Rssia).. Abdshrov A.A. Estimators of ow distribtios from icomplete observatios ad its properties. LAMBERT Academic Pblishig p. (I Rssia). 3. Abdshrov A.A. Nrmhamedova N.S. Approimatio of the Lielihood Ratio Statistics i competig riss model. // Doclady Acad. Na. Uzbeista. 0. N.3. (I Rssia to appear). 4. Abdshrov A.A. Nrmhamedova N.S. Approimatio of the lielihood ratio statistics i competig riss model der radom cesorship from both sides // ACTA NUUz. 0. N.4. p.6-7. (I Rssia). 5. Bobooov J.A. Nrmhamedova N.S. // Approimatio of the lielihood ratio statistics i competig riss model der radom cesorship from the right // I Statistical Methods of Estimatio ad Hyphoteses Testig. Perm State Uiversity. Perm. 0 Isse 3 p (I Rssia) 6. Wald A. Tests of statistical hypothesis cocerig several parameters whe the mber of observatios is large// Tras. Amer. Math. Soc p Le Cam L. O some asymptotic properties of the maimm lielihood estimates ad related Bayes estimates.// Uiv. Califoria Pbl. Statist v.. p Hae J. Local asymptotic miima ad admissibility i estimatio // Proc. Sith. Bereley Symp. o Math. Statist. ad Prob V.. -P Rsas J. Cotigity of probability measres. М.: Мir p. (I Rssia). 0. Ibragimov I.A Khas'misii R.. Asymptotic estimatio theory. М.: Naa p. (I Rssia).. Va der Vaart A.W. Asymptotic Statistics. Cambridge Uiv. Press p. 64 Pa..stat.oper.res. Vol.XII No. 06 pp55-64