Iterated Logarithm Laws on GLM Randomly Censored with Random Regressors and Incomplete Information #

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1 Appled Mathematcs do:.436/am..343 ublshed Ole March ( Iterated Logarthm Laws o GLM Radomly Cored wth Radom Regrsors ad Icomplete Iformato # Abstract Qag Zhu Zhhog ao Guagla Q Fag Yg Statstcs Rearch Isttute Huahog Agrcultural Uversty Wuha Cha School of Scece Hube Uversty of echology Wuha Cha E-mal: hhsfym@mal.hau.edu.c Receved December 8 ; revsed Jauary 6 ; accepted Jauary 9 I ths paper we defe the geeraled lear models (GLM) based o the observed data wth complete formato ad radom corshp uder the case that the regrsors are stochastc. Uder the gve codtos we obta a law of terated logarthm ad a Chug type law of terated logarthm for the mamum lkelhood tmator (MLE) ˆ the pret model. Keywords: Geeraled Lear Model Icomplete Iformato Stochastc Regrsor Iterated Logarthm Laws. Itroducto Corrpodg Author. #Foudato tems: supported by Huahog Agrcultural Uversty Doctoral Fud (54-67) ad Iterdscplary Fud (8 kjc8 ad orch la Fud (9H3). he geeraled lear model(glm) was put forward by Nelder ad Wedderbur [] 97s ad has bee studed wdely sce the. he mamum lkelhood tmator (MLE) ˆ of the parameter vector GLM was gve ad ts strog cosstecy ad asymptotc ormalty were dscussed by Fahrmer ad Kaufma [] 985.he radomly cored model wth the complete formato was preted by Elper ad Gertsbak [3] 988.he aalyss of the radomly cored data wth complete formato has become a ew brach of the Mathematcal Statstcs. ao ad Lu [4] 8 dscussed the strog cosstecy ad the asymptotc ormalty of MLE ˆ of GLM based o the data wth radom corshp ad complete formato. ao ad Lu [5] dscussed laws of terated logarthm for quas-mamum lkelhood tmator of GLM 8 meawhle ao ad Lu [6] 9 dscussed laws of terated logarthm for mamum lkelhood tmator of geeraled lear model radomly cored wth complete formato uder the regrsors gve. However La ad We [8] Zeger ad Karm [9] have studed the lear regrso model uder the case that the regrsors are stochastc. I the practcal applcato pecally the bomedcal socal scec the regrsors GLM are ofte stochastc Fahrmer [] vtgated GLM wth the regrsors whch are depedet ad detcally dstrbuted ad gave MLE of matr parameter wthout proof uder the gve codtos. Dg ad Che [] 6 gave asymptotc propert of MLE GLM wth stochastc regrsors. So the pret paper we wll vtgate the law of terated logarthm ad the Chug type law of terated logarthm for mamum lkelhood tmator of geeraled lear model radomly cored wth complete formato uder the case that regrsve varabl are depedet but ot ecsarly detcally dstrbuted. From a statstcal perspectve the mportace of those laws stem from the fact that the frst oe gv a asymptotc se the smallt % cofdece terval for the parameter whle the secod oe gv a almost sure lower boud o the accuracy that the tmator ca acheve.. Model wth the Radom Regrsor Suppose that the rpodece varabl Y are oe dmo radom varabl ad regrsor varable are q-dmo radom varabl whch have the dstrbuto fuctos K rpectvely. Here s the observato value of. Wrte. Suppose that the observatos Y are mutually depedet ad satsfy. Copyrght ScR.

2 364 ) he regrso equato: EY m (.) q where the ukow parameter B. ) he codtoal dstrbuto of Y uder s the epoet dstrbuto.e. Y dy C y ep yb dy (.) where s a -fte measure parameter : Cyepydy s the atural parameter space ad s the teror of. Sce ths codtoal dty tegrat to we see that b Cepydy from whch the stadard eprsos for the codtoal mea EY b Var Y ad the varace b b where b deote the frst ad secod dervatv of b rpectvely. Suppose that the cor radom varabl U are mutually depedet but ot ecsarly detcally dstrbuted wth the dstrbuto fucto dg u g u du K d G u ad. Deote d. Suppose that U s depedet of Y For let IY U f Y U but the real va lue of Y s't observed else Yf Z U otherwse Obvously Z s a mutually depedet ad observable sample. he codtoal dty ad dstrbuto fucto of Y uder are rpectvely deoted as f y; Cyep y b ; ep F C y y b d y Y Let G G Q. ZHU E AL. ; F ; F. Suppose Y y U u p f y u (.3) Y y U u p (.4) f y u where p. hs assumpto came from. Elper ad I. Gertsbak [3]. I the relablty study the stat of a tem's falure s observed f t occurs before a radomly chose specto tme ad the falure s sgaled. Otherwse the epermet s termated at the stat of specto durg whch the true state of the tem s dscovered.. Elper ad I. Gertsbak assumed that the falure tme of every tem was sgaled radomly wth probablty p before the radomly chose specto tme. he we have Y yu u s ds- Y y U u y u Wthout loss of geeralty assumg that crete we have Y yu u Y y U u (.5) We frst gve the followg propostos. roposto.. Uder the regular assumptos above we have Z (.6) p G y f y dy Z (.7) p F y; dgy Z (.8) Fy ; dg y. roof. We oly show (.6) for the dscrete case the cotuous case ca be show the way smlar to that of the dscrete case. Z EI I E I Y U Y Y U y Y yu u YdyU du p G y df y p G y f y dy (.9) Copyrght ScR.

3 Q. ZHU E AL. 365 where (.9) follows from (.3) ad (.5). Smlarly we ca demostrate (.7) ad (.8). Suppose that s the observato of Z s the observato of s the observato of (.6) (.7) ad (.8) mply that for all the codtoal dstrbuto of Z uder s the followg pg( ) f( ; )] [( p) F( ; ) g( ) ; F g d Let ( ) (.). Z Z Z We easly get the followg proposto. ad roposto.. For all we have Z Z Z (.) Z Z Z ( ) (.) where Z( ) ( ) meas Z for. Remark.. roposto. mpl that uder U are mutually depedet ad so are Y ad Z. (.) ad (.) mply that the codtoal dstrbuto of Z Z uder s pg ; ; ; f p F g F g d (.3) he codtoal probablty measure corrpodg to (.3) s wrtte as. Meawhle let E ad Var deote the codtoal epectato ad codtoal varace uder the codtoal probablty measure rpectvely. Set do- ote the real value of. For otatoal smplcty let ad Var E E Var. (.3) mpl that the jot dstrbuto of s Z Z pg ; ; ; f p F g F g d d (.4) he probablty measure (ucodtoal) corrpodg to (.4) s deoted as. Meawhle let E ad Var deote the epectato ad varace uder the probablty measure rpectvely. For otatoal smplcty let ad Var E E Var It s that the parameters (.4) are studed by us. 3. Ma Rults Furthermore from (.4) we get the lkelhood fucto of Z Z as follows L Z Z pg Z f Z; p F Z; g Z (3.) FZ; g Z akg the logarthm to (3.) ad droppg the terms whch are free of yeld the logarthm lkelhood fucto: log ; log FZ; log FZ; l f Z l ; Z Z where l ; Z Z (3.) s the logarthm lkelhood fucto defed ao ad Lu [8]. Copyrght ScR.

4 366 Q. ZHU E AL. We have the score fucto l Z b yf y dy Z F Z; F Z ; Z ; yf y; dy ; Z Z (3.3) where ; Z Z s defed as ao ad Lu [8]. Ad l H b yf y dy Z = ; F Z; H F Z F Z ; Z ; ; y f y dy yf Z y dy ; ; F Z ; y f Z y dy ; Z Z where H ; Z Z s defed as ao ad Lu [8]. Wrte ; E E H; where ; s defed as ao ad Lu [8]. he soluto of the logarthm lkelhood equato (3.4) s wrtte as Z Z. (3.5) (3.3) ad (3.4) mply that ˆ Z Z (3.6) where ˆ Z Z s defed as ao ad Lu [8]. he orm of matr A aj s pq p q defed as A j a j. We wrte as the usual er product ad e s as the sth caocal bass q. Let F yf y dy ; F y f y dy ; F yf y dy 3 ; F y f y dy 4 ;. We state the followg assumptos: ( C ) For all for all B a.s. where. Here s compact. ( Q ; lm ; s a q-order ad E C ) postve defe matr. ( C 3 ) For all B ; ; L (3.7) L L L ; ; (3.8) ; ; (3.9) ; ; (3.) where b sup ELj Z; Lj as.. j 34 b. ( C 4 ) B E t a.s. It s also easy to see that the codtos the pret paper mply the codtos (C) (C) (C3) ad (C4) gve ao ad Lu[8]. So there almost sure ests the mamum lkelhood tmator of. Hece our frst rult stat a law of the terated logarthm for the mamum lkelhood tmator of. heorem 3.. Uder codtos ( C ) ( C ) ( C 3 ) ad ( C 4 ) f ˆ s the MLE of the for s q we have Copyrght ScR.

5 Q. ZHU E AL. 367 ad ˆ lmsup < Q as.. loglog ˆ lmf < Q as.. loglog roof. For arbtrarly gve we regard the codtoal probablty measure as the probablty measure defed ao ad Lu [8] ad ote that as s gve MLE s equvalet to MLE ˆ ˆ Z Z obtaed ao ad Lu [6]. hus Remark. mpl heorem. ao ad Lu [8] ad hece we have the dred rults. Remark 3.. Uder the codtos of heorem 3. we take epectatos for the rults above ad mmedately get ad ˆ lm sup < Q loglog ˆ lm f < Q loglog Note that heorem 3. tablsh a law of terated logarthm for each compoet of ˆ. Our et rult s a Chug type law of terated logarthm. o ths am we add ad addtoal codto. For otatoal smplcty let s Qt. he s eq t Q e s s We make the followg assumpto: C ) ( 5 f E s k ki a.s. where I E s : ( ) a.s.. heorem 3.. Uder codtos ( C ) ( C )( C 3 ) ( C 4 ) ad ( C 5 ) f ˆ s the MLE of the for s q we have log log lmf ma ˆ Q( ) as.. 8 roof. I the way smlar to that of heorem 3. we mmedately obta the dred rult. Remark 3.. Uder the codtos of heorem 3. we take epectatos for the rults above ad mmedately get log log lmf ma ˆ Q( ) 8 4. Coclusos he rults obtaed the pret paper are based o the case that the lk fucto s a atural lk fucto. However Dg ad Che [9] gave the cosstecy ad asymptotc ormalty of MLE ˆ of GLM uder the case that the lk fucto s of o-atural lk hece the academcas who are terted GLM may furthermore vtgate the terated logarthm law ad Chug type terated logarthm law of MLE ˆ of GLM uder the case that the lk fucto s of o-atural lk. 5. Ackowledgemets he authors would lke to thaks the ukow refere for helpful commets. 6. Referec [] J. A. Nelder ad R. W. Wedderbur Geeraled Lear Models Joural of the Royal Statstcal Socety: Ser Copyrght ScR.

6 368 Q. ZHU E AL. A 35 art 3 97 pp do:.37/34464 [] L. Fahrmer ad H. Kaufma Cosstecy ad Asymtotc Normalty of the Mamum Lkelhood Estmator Geeraled Lear Models Aals of Statstcs Vol. 3 No. 985 pp do:.4/aos/ [3]. Elper ad I. Gertsbak Estmato a Radom Corg Model wth Icomplete Iformato: Epoetal Lfetme Dstrbuto IEEE rasactos o Relablty Vol. 37 No. 988 pp do:.9/ [4]. L. La ad C. Z. We Least Squar Estmat Stochastc Regrso Models wth Applcatos to Idetfcato ad Cotrol of Dyamc Systems Aals of Statstcs Vol. No. 98 pp do:.4/aos/ [5] S. L. Zeger ad M. R. Karm Geeraled Lear Models wth Radom Effects; A Gbbs Samplg Approach Joural of the Amerca Statstcal Assocato Vol. 86 No pp do:.37/8977 [6] Z. H. ao ad L. Q. Lu MLE of Geeraled Lear Model Radomly Cored wth Icomplete Iformato Acta Mathematca Sceta Vol. 3(A) 8 pp [7] Z. H. ao ad L. Q. Lu. Laws of Iterated Logarthm for Quas-Mamum Lkelhood Estmator Geeraled Lear Model Joural of statstcal plag ad ferece Vol. 38 No. 3 8 pp do:.6/j.jsp.6..6 [8] Z. H. ao ad L. Q. Lu Laws of Iterated Logarthm for MLE of Geeraled Lear Model Radomly Cored wth Icomplete Iformato Statstcs ad robablty Letters Vol. 79 No. 6 9 pp do:.6/j.spl.8..6 [9] J. L. Dg ad. R. Che Asymptotc ropert of the Mamum Lkelhood Estmate Geeraled Lear Models wth Stochastc Regrsors Acta Mathematca Sca Vol. No. 6 6 pp do:.7/s Copyrght ScR.