ASYMPTOTICS OF THE GENERALIZED STATISTICS FOR TESTING THE HYPOTHESIS UNDER RANDOM CENSORING

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1 IJRRAS 3 () Novembe ASYMPOICS OF HE GENERALIZE SAISICS FOR ESING HE HYPOHESIS UNER RANOM CENSORING A.A. Abduhukuov & N.S. Numuhamedova Natoal Uvety of Uzbekta ahket ABSRAC I th atcle a model of adom ceohp fom both de we code eeach two tattc fo tetg the compote hypothee whch have lmt a ch-quae dtbuto wth appopate degee of feedom. Ft oe the geealzed ch-quae tattc fo the cotucto of whch we ue the powe etmate dtbuto of fucto(d.f.). he ecod tattc twce the logathm of the lkelhood ato tattc (LRS) of model of adom ceohp fom both de. Both of thee tattc ca be ued to cotuct a aymptotc tet of ch-quae type fo the compote hypothee. Keywod: ch-quae tattc lkelhood ato tattc maxmum lkelhood etmate adom ceog.. INROUCION Let {( X Y Y ) } equece of depedet ad detcally dtbuted (..d) adom vecto wth mutually depedet compoet ad magal d.f.- F ad G k fo adom vaable (.v.) X ad Y k k ; epectvely. Code the cae whe.v. X ubect to adom ceog fom both de by vaable Y k. O - th tage of the expemet we obeve the ample of ze : Z Y X Y ( ) I X Y Y ( ) S {( Z ) } ( ) I( Y X Y ) I( Y Y X ). Hee fo umbe a ad b : a b m( a b) a b max( a b). I a ample S.v. X obeved oly whe. I th model of adom ceohp fom the both de of the poblem cot etmatg of codtoal uvval fucto ( ) F ( x) P( X x / X ) x fom ample S ude uace pa ( G G ) fo pecfc umbe. I th atcle we code the poblem of tetg the compote hypothe H : FF F { F ( ; ) } - ( ) famly of dtbuto deped o ukow paamete (... ) ad - a ope et R. Code two tattcal tet fo vefy H wth a lmt of ch-quae dtbuto.. GENERALIZE CHI-SQUARE SAISICS Fo buld tattc of ch-quae tet we code the opaametc etmate of F ( x) fom []: R ( x; ) ( x; )( L ( x; )) R ( x; ) q ( x) F ( x) x q () () ( x; ) ( q ( u)) dh ( u) [ ; x] L ( x; ) ( q ( u)) dq ( u) q ( x) G ( x) H( x) [ ; x] () () () x H H ( x) I( Z x) H ( ) H ( x) ( x) ( m) m H ( x) I( Z x) m 567

2 IJRRAS 3 () Novembe Abduhukuov & Numuhamedova Hypothe Ude Radom Ceog () G ( x) exp H ( u) dh ( u) x. [ x; ) I ode to cotuct tet tattc we toduce the codto (C) Let d.f.- F ad G cotuou ad the umbe uch that ad f P ( Y x X Y ) ; x (C) Suppot N { x : F( x; ) } depedet o ; F (C3) hee a dety f( x; ) wth d.f. Fx ( ; ) t ha cotuou devatve: f( x; ) dx ; ; f( x; ) ad (C4) Ifomato matx of Fhe I ( ) I potve defte ad cotuou by log f ( x; ) log F( x; ) I ( ) ( G ( x) G ( x)) df( x; ) F( x; ) dg ( x) log( Fx ( ; )) ( F ( x ; )) dg ( x ) ; (C5) hee a maxmum lkelhood etmate (MLE) (... ) fo paamete (... ) obtaed by olvg the ytem of equato log p ( )... p ( ) ( F( Z; )) ( f ( Z; )) ( F( Z; )) - the tucated lkelhood fucto of the model. Moeove the MLE ca be epeeted by / ( ) I ( ) A ( ) o () p / log p ( ) A ( ) omalzed cotbuto fucto. Remak. It hould be oted that the codto (C) ad (C3) eue the extece of ecod-ode devatve of fucto log f( x; ) log Fx ( ; ) ad log( Fx ( ; )). Ideed fo all ad ( x; ) NF : 568

3 IJRRAS 3 () Novembe Abduhukuov & Numuhamedova Hypothe Ude Radom Ceog log f ( x; ) f ( x; ) f ( x; ) f ( x; ) f( x; ) f ( x; ) log F( x; ) f ( x; ) f ( x; ) f ( x; ) dx dx dx Fx ( ; ) F ( x; ) log( F( x; )) f ( x; ) dx Fx ( ; ) f ( x; ) f ( x; ) dx dx ( Fx ( ; )). We peet the aymptotc popete of etmate / F fom []. We defe a equece of pocee V ( x) ( F ( x) F ( x)) x. Fo thee pocee the equece of appoxmatg pocee M ( x) ( F ( x)) N ( x) x () () ( B ( u) ( u)) dh ( u) B ( x) ( G ( u) H( u )) G x N ( x) ( ) H( x) () x () B ( ) B ( u) d( G ( u) H( u)) ( ) ( ) ( ( ) ( )) G N G u H u ( x) G ( x) () () () B ( u) dh ( u) B ( x) B ( u) dh ( u) H ( u) H( x) H ( u). m Hee fo each : H( x) P Z x EH ( x) ( m) ( m) B ( u) B( H ( u)) m ad B( y) y ( ) ( m) m H ( x) P Z x EH ( x) ; B ( u) B( H( u)) poce of a Bowa bdge. Note that the pocee M ( x ) ae lea fuctoal of the Bowa bdge ad thu ae Gaua pocee wth zeo mea. We peet the followg theoem fom [ heoem..]. heoem A. []. Ude codto (C) we have a appoxmato / P up V( x) M( x) R log Q x 569

4 IJRRAS 3 () Novembe Abduhukuov & Numuhamedova Hypothe Ude Radom Ceog R R( ) ad Q (abolute) potve cotat. Remak. I codto of heoem A fo by lemma of Boel-Catell we have the tog appoxmato a.. / up V ( x) M ( x) O log. x Fom hee we have the weak covegece V ( x) M ( x) [ ; ] () M ( ) M( ) fo each ad Gaua poce M( x ) obtaed fom M ( x ) by eplacemet of B ( u ) ad ( m B ) ( u ) m by the appopate Bowa bdge wth agumet Hu ( ) ad epectvely. We toduce the adom pocee x x ( m H ) ( u) m / x x ( x; ) ( F ( x) F ( x; )) o x... x poble adom patto fo a gve pobablty F( t ; ) p. Code a adom vecto ( x )... ( x ). he ext eult geealze (). heoem. Let fo all the codto (C)-(C5) hold. he the adom poce covege weakly to the Gaua poce ( x ) wth zeo mea ad covaace wth x y : F( x; ) F( x; ) F( x; ).... F( x; ) F( y; ) Cov ( x) ( y) Cov M ( x) M ( y) ( ) I ( ) ( ; ). Let p atfyg the equalty Poof of the heoem. Expad the poce ( x ) the eghbohood of ad the we have ( x ) x ( x ) V ( x ) V ( x ) R ( x ) () Fx ( ; ) / V ( ) x ad ude fom heoem A up R ( x) o (). x p 57

5 IJRRAS 3 () Novembe Abduhukuov & Numuhamedova Hypothe Ude Radom Ceog Code a fte-dmeoal dtbuto of um let V x V x x... x x / V B B B matx wth elemet B (C5) ad multvaate cetal lmt theoem V ( x) V ( x). Fo a abtay patto ( )... ( ) V ad V x V x V ( )... ( ) uch that Ft ( ; ). By () codto ( ) ( ) M= Cov M x M x / V V N(; ) (3) M I I ( ) vee matx of ( ) I. hu ; V V M B N. Ude codto (C)-(C5) fom eult of [3] follow that ad.v. depedet ad I theefoe M= B B. Hece we have M M B I B. (4) I vew of ()-( 4) ug the techque of poof of heoem 4 [] we ee that the weak covegece of the um V ( x) V ( x) follow fom the weak covegece to the cotuou lmt of dvdual ummad. Covegece V ( x ) follow fom () poce V ( x ) cot of poduct of o-adom fucto to a aymptotcally omal equece of adom vaable ad theefoe t covegece to the lmt obvou. h complete the poof of heoem. fucto Let M ad etmate of the matce M= ad obtaed by eplacg o MLE. he ( ) m G H ad H eplaced by the opaametc etmate G geeal pcple of cotucto of ch-quae tattc (ee [45]) we code the tattc ( m) H ad H m. Followg the ( x )... ( x ). he we have heoem. Let the codto (C)-(C5) hold ad ag. he L / H K 57

6 IJRRAS 3 () Novembe Abduhukuov & Numuhamedova Hypothe Ude Radom Ceog K ch-quae dtbuto wth degee of feedom. 3. CHI-SQUARE ES BASE ON HE LIKELIHOO RAIO SAISICS Ft let cala paamete. Code a mple hypothe H : agat the compote alteatve H: { }. Let MLE atfyg the codto (C5) ad code LRS L p. p We alo code the codto (C6) hee a thd devatve o of dety f( x; ) ext the depedet of fucto h uch that d 3 log p ( ) d 3 h ; M h. Fom the geeal theoy of MLE (ee [6]) follow that ude codto (C) - (C4) (C6) thee ext a uque cotet MLE ad at / L / H N I ( ). he by aylo fomula log p ( ) log L log ( ) log ( p p ) 3 log p( ) log p( ) 3 3! 3 (5) ad 3 log p ( ) log p ( ) log p ( ) log p ( ) 3 log p (. Sce ) the ubttutg the expeo fo (6) to (5) we have log p ( ) (6) fom 57

7 IJRRAS 3 () Novembe Abduhukuov & Numuhamedova Hypothe Ude Radom Ceog log L log p ( ) q q o () at. Now ug the low of lage umbe ad cetal lmt theoem we fd that ude p hypothe H at tattc log p ( ) I ( ) ( I ( )) have a ch-quae dtbuto K wth oe degee of feedom.e. hu we have poved heoem 3. Ude codto (C) - (C6) ad the hypothe H L log L / H K. (7) log L. (8) heoem ad 3 ca be ued to cotuct a aymptotc ch-quae tet fo the hypothee H. Remak 3. It hould be oted that the cae dmeoal paamete (... ) the lmt (7) we wll have a ch-quae dtbuto K.e. tead of (8) we have log L covegece (7) ug a multvaate cetal lmt theoem fo tattc log p( ) log p( ) q log L ( ) ( ). h eult poved mlaly to q o (). 4. REFERENCES p []. Abduhukuov A.A. Etmato of ukow dtbuto fom complete obevato ad t popete. LAMBER Academc Publhg.. 99 p. (I Rua). []. Below N. Cowley J. A lage ample tudy of the lfe table ad poduct lmt etmate ude adom ceohp.// A. Statt v.. p [3]. Pece.A. he aymptotc effect of ubttutg etmato fo paamete ceta type of tattc. // A. Statt. 98. v.. p [4]. Voov V.G. Nkul M.S. Ubaed etmate ad thee applcato. М.: Scece p. (I Rua). [5]. Voov V. Naumov A. Pya N. Some ecet advace ch-quaed tetg. // Poceedg of Iteat. Cof. Advace Stattcal Ifeetal Method : Almaty. Kazakhta. 3. p [6]. Zack Sh. he theoy oy tattcal feece Joh Wley & So Ic. New Yok-Lodo-Sydey-ooto p. 573