有序参数的一类AIC型信息准则_英文_

有序参数的一类AIC型信息准则_英文_ 2O n an A IC typ e In fo rm a t io n C r ite r io n Ξ fo r O rde red P a ram e te r s C hen D o ng ( ), , 100101, D ep a r tm en t of B as ic C ou rseB e ij ing U n ion U n iv e rs ity B e ij ing C h ina () : , A bstrac tIn th is p ap e rA new m e tho d ba sed o n th e p r inc ip le o f th e A k a ik e Info rm a t io n C r ite r io n A IC is . , g ivenT h e m e tho d p ro v ide s an exac t ly unb ia sed e st im a to r o f th e b ia s co r rec t io n te rm o f th e A IC unde r th e . , a ssum p t io n o f th e no rm a l d ist r ibu t io n s fo r th e o rde r re st r ic ted p a ram e te r sF u r th e ra m e tho d com b ined w ith .th e boo t st rap sam p ling is a lso deve lop ed () : , , .Key wordsA k a ik e info rm a t io n c r ite r io n A IC Boo t st rap sam p lingP a ram e te r e st im a t io n : 212. 1 : CL C n um berO D ocum en t codeA 0 B ack g ro u n d ( ) ?, , . Ηi ?i denSuppo se th a t in dep en den t ran dom sam p le s f rom k pop u la t io n sL e t f o te th e den sity 2= 1, 2, , . , , fo r th e ith pop u la t io n w h e re ?i is k now n an d Ηi is u n k now n fo r ikIn m an y situ a t io n ce r ta in Φ Φ Φ . . o rde r re st r ic t io n m ay b e a ssum ed o n ΗsT h e sim p le o rde r re st r ic t io n Η1 Η2 Ηk is a typ ica l o n eIn 2, b io a ssay s o f do seeffec t exp e r im en t sit w ill b e suppo sed th a t th e to x ic effec t s o r adve r se effec t s o f .ce r ta in d ru g s o r ch em ica ls w o u ld in c rea se w ith th e do sage leve ls , . , Fo r th e eva lu a t io n o f th e effec t iven e ss o f d ru g ssta t ist ica l te st s m ay b e rea so n ab leH ow eve rfo r , 0.n ega t ive effec t su su a l sta t ist ica l te st s a re no t app rop r ia te if th e sign if ican ce leve l is se t a t u su a l va lu e 05 . , . , o r soO b v io u slysu ch n ega t ive effec t sho u ld b e de tec ted sen sit ive lyU n fo r tu n a te lysta t ist ica l te st s , w ill b e du ll fo r de tec t in g su ch n ega t ive effec t s a t u su a l sign if ican ce leve lssin ce sta t ist ica l te st s a re .suppo sed to k eep th e p ro b ab ility o f th e typ e I e r ro r a t o r u n de r th e sign if ican ce leve ls ( ) , , 1993 Fo r th is p ro b lem K ik u ch iY an agaw a an d N ish iyam a p ropo sed a m e tho d b a sed o n th e () , , A k a ik e In fo rm a t io n C r ite r io n A IC a s an a lte rn a t ive to sta t ist ica l te st ssp ec if ica llyfo r de tec t in g th e . m ax im a l do se leve l a t w h ich th e in c rea se o f th e adve r se effec t is rega rded a s accep tab ly sm a llT h e ( ) , 1997m e tho d w a s deve lop ed an d app lied to va r io u s p ro b lem s in Y an agaw aK ik u ch i an d B row n an d ( ) 2001, Y an agaw a an d K ik u ch i w h e re th e n um b e r o f d ist in c t va lu e s in th e o rde r re st r ic ted m ax im um () lik e lihoo d e st im a te s M L E s o f th e m ean s w e re em p lo yed a s th e p en a lty te rm fo r sp ec if ic m o de ls .reco n st ru c ted b a sed o n th e M L E s fo r th e m ean s , . In th is p ap e rW e p ropo se a n ew m e tho d b a sed o n th e p r in c ip le o f th e A ICT h e m e tho d p ro v ide s an , ex ac t ly u n b ia sed e st im a to r o f th e b ia s co r rec t io n te rm o f th e A IC u n de r th e a ssum p t io n o f th e no rm a l . , d ist r ib u t io n s fo r th e o rde r re st r ic ted p a ram e te r sF u r th e ra m e tho d com b in ed w ith th e boo t st rap .sam p lin g is a lso deve lop ed 1 In t ro du c t io n ( ) L e t X deno te th e o b se rva t io n o f a ran dom sam p le f rom a pop u la t io n w ith den sity g ?. Fo r th is, o b se rva t io n w e m ay a ssum e a c la ss o f den sity fu n c t io n s de sc r ib ed b y an u n k now n P a ram e te r Η an d a δ ( ), ?, , . . k now n p a ram e te r ?th a t is f Η?A n d le t xΗ deno te an e st im a te o f ΗT o m ea su re th e d istan ce δ () () ??, , 2x b e tw een th e t ru e den sity g an d th e e st im a ted den sity f Η?K u llb ack L e ib le r d istan ce δ δ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ()d [ g , f t, Η, ?= g tlo g [ g td Λt- g tlo g [ f t, Η, ?d Λt1x x ?? () () ( () ) is em p lo yed see K u llb ack an d L e ib le r 1951, p ro v ided g ?, f ?, Η, ?is th e den sity w ith re sp ec t to . , 2Ρf in ite m ea su re ΛA m o n g seve ra l can d ida te e st im a te s o f th e t ru e den sityth e o n e w h ich g ive s th e la rge r va lu e o f δ ( ) ( ) ( ) g tlo g [ f t, Η, ?d Λtx ? δ) ( . , 2x a s th e e st im a te m ay b e p refe rab le to th e o th e r sH ow eve rif w e em p lo y th e lo glik e lihoo d H Η δ ( ) ( ) ( ) g tlo g [ f t, Ηd Λt,x ? :it y ie ld s a b ia s in th e fo llow in g sen se δ δ () () ( ) ( ) ( ) ()B Η= E {L Η- g tlo g [ f t, Η, ?d Λt}. 2x x ? () ?. , W h e re th e exp ec ta t io n is tak en u n de r th e t ru e den sity g T h u sto com p a re d iffe ren t m o de ls w ith Ηs δ ( ) , 22- o f d iffe ren t d im en sio n s o r reg io n sth e b ia sco r rec ted lo glik e lihoo d fo r th e re sp ec t ive m o de l L Ηx B () .Ηw o u ld b e u sed fo r th e com p a r iso n ( ) ( ), ??, , , . In th e A IC th e t ru e den sity g is co n side red to b e equ a l to som e f Η?M o reo ve rth e δ() , , . u n re st r ic ted m ax im um lik e lihoo d e st im a te M L E deno t in g Ηis em p lo yedU n de r p rop e r regu la r ity , co n d it io n sth e d ist r ib u t io n o f δδ() ( ) ( ) ( )L Η- g tlo g [ f t, Ηd Λt ? w ill b e a sym p to t ica lly app ro x im a ted b y a ch i squ a red d ist r ib u t io n w ho se deg ree s o f f reedom equ a ls th e ( ( ) ( )1973, 1974, 1976.n um b e r o f u n k now n p a ram e te r s o f th e t ru e d ist r ib u t io n see A k a ik e T ak eeu ch i , T h u sth e A IC is def in ed b y δδδδ() () ( ) ( ) ( ) () ()A IC Η= L Η- g tlo g [ f t, Ηd Λt> L Η- p. 3 ? δ() - 2 - .N o te th a t A IC is u su a lly def in ed b y L Ηp ( ) , ?, , , T o sim p lify th e d ist r ib u t io n w e a ssum e th a t f Η?is a no rm a l d ist r ib u t io n w h e re Η is th e () () ( , ?, , , a ltho um ean an d g Ηis equ a l to som e f Η? gh seve ra l e st im a to r s m ay b e po ssib le to choo se see () ) 1996, , .Ko n ish i an d K itagaw a an d th en w e em p lo y th e m ax im um lik e lihoo d p r in c ip lein th is p ap e r , = 1, 2, , , = 1, , In co n side r in g k o b se rva t io n s x i j i k j n i o f in dep en den t ran dom sam p le s f rom k () ?, , , = 1, 2, , , no rm a l d ist r ib u t io n s f Ηi ?i w ith u n k now n m ean Ηi an d k now n va r ian ce T i fo r ik sim ila r :d iscu ssio n lead s to th e fo llow in g b ia s te rm k δ( ) ( ) f t, Η, ?lo g [ f t, Η, ?d t.() () i i i i i i i B Η= E L Η- n Η i ? ?i= 1 , . , If no sp ec if ic o rde r re st r ic t io n is a ssum ed o n u n k now n p a ram e te r sit lead s to th e u su a l A ICH ow eve r () (() ), 1999.if ce r ta in o rde r re st r ic t io n is a ssum edB Ηdep en d s o n ΗA n rak u , Gen e ra llyit is cum b e r som e to eva lu a te th e d ist r ib u t io n s o f th e o rde r re st r ic ted m ax im um lik e lihoo d (( ) )1988. , T h u s, e st im a to r an d th e ir fu n c t io n s even fo r a sim p le ca se R o b e r t so n W r igh t an d D yk st ra ( )in stead o f seek in g th e b ia s te rm ex ac t ly, w e co n side r th e m e tho d o f e st im a t in g B Η. T h e ca se s w h e re .th e va r ian ce s in vo lve a comm o n u n k now n p a ram e te r w ill b e a lso d iscu ssed 2 A n in fo rm a t io n c r ite r io n fo r th e ca se o f no rm a l d is t r ib u t io n s , , , , Fo r k no rm a l d ist r ib u t io n s w ith m ean s Η1 Η2 Ηk som e o f o rde r re st r ic ted re la t io n s a re g iven a s :fo llow s () ( ) 1Φ Φ Φ ;Η1 Η2 Ηk sim p le o rde r () (() ) ; 2ϖ 1Φ Φ - 1, . . Φ Φ Ε Ε j j k stΗ1 Ηj Ηj + 1 Ηk u n im o da l o rde r () ( ) ( )3Φ = 2, , .Η1 Ηi ik sim p le t ree o rde r k ) ( T h e se t o f Η= Η1 , Η2 , , Ηk ′sa t isfy in g ce r ta in o rde r re st r ic t io n is a po lyh ed ra l co n e in R, an d th e . .co n e is co n vexD eno te th e po lyh ed ra l co n e b y C δ Η=T h e u n re st r ic ted m ax im um lik e lihoo d e st im a to r o f Ηi is g iven b y th e co r re spo n d in g sam p le m ean i n - 1 i kδ δ δ δ (n ) x . = , , , ′, , ?, >, < i i jSe t ΗΗ1 Η2 Ηk an d fo r an y x y R def in e th e in n e r p ro du c t o f th em b y x y ?j = 1 () = x ′y , p ro v ided + = d iag n 1 ƒ?1 , n 2 ƒ?2 , , n k ƒ?k . L e t ‖?‖ deno te th e in n e r p ro du c t no rm. M o reo ve r, υ υ υ υυ υ υ ) ( 1 2 , , , k , , , , , = , , , .′le t ΗΗΗdeno te th e m ax im um lik e lihoo d e st im a to r o f Η1 Η2 Ηk an d se t ΗΗ1 2 ΗΗk υ δ . T h en th e vec to r Ηis th e p ro jec t io n o f Ηo n to C w ith th e no rmN o te th a t th e p ro jec t io n m ean s th e . o r tho go n a l p ro jec t io nSuppo se w e h ave in dep en den t ran dom sam p le s x i j f rom a no rm a l d ist r ib u t io n w ith υυ , , = 1, 2, , , = 1, , . 2i u n k now n m ean Ηan d a k now n va r ian ce ?i fo r ik j n iT h en th e lo glik e lihoo d a t Ηis n k ki 2υ() ij i x - ΗN1 1 υ () ) (. L Η= - lo g 2Π- n lo g?- i i ???2 2 2 ? ii= 1i= 1 j = 1 k N = n . , , 2P ro v ided i T h e refo refo r f ix ed Ηth e exp ec ted lo glik e lihoo d is ?i= 1 kkk2υ () n Η- Η i N i 1 1 υυ ) (( ) ( ) l o g 2Π+ 1 -nf t, Η, ?lo g [ f t, Η, ?]d t= -n lo g ?-.ii i i i i i i i i ????2 2 2 ? i i= 1 i= 1i= 1 υ() 2: , T h u sif w e e st im a te th is b y th e lo glik e lihoo d L Η, th e b ia s is rep re sen ted a s fo llow s kυυυ( ) ( ) () () B Η= E L Η- n f ti , Ηi , ?i lo g [ f ti , Η, ?i ]d tii Η i ? ?i= 1 k 2 υ2 nk (Η - Η - Η - Η . i = + E Η ? i 2 i= 1 i υδ ) ( ) i i 2? i υ() , U sin g no rm sth e b ia s te rm B Ηis exp re ssed a s 1 2 2 2 δ δ υυ () B Η= E Η ‖Η- Η‖- ‖Η- Η‖+ ‖Η- Η‖.2 S in ce δ δ δ δ 2 2 2 2 υυ υ υ () () ‖Η- Η‖= ‖Η- Η- Η= ‖Η- Η‖+ ‖Η- Η‖- 2 < Η- Η, Η- Η> ,- Η‖ υ() < , - > = 0, , an d E ΗΗΗΗCo n sequ en t lyw e h ave υυ () ()() B Η= E < Η, Η- Η> . 4Η () , , In gen e ra lth e b ia s te rm B Ηdep en d s o n th e u n k now n p a ram e te r Ηan d it seem s cum b e r som e to o b ta in ( ). , , th e va lu e an a ly t ica llyT h u sin stead o f seek in g th e b ia s te rm ex ac t lyw e e st im a te th e b ia s te rm B Η .f rom th e da ta = 1, 2, , , Fo r ik suppo se th a t Y i is a no rm a lly d ist r ib u ted ran dom va r iab le w ith m ean Ηian d va r ian ce υ υ υ υ υ υ () . , , , , , , , , , , , ƒ?i n iL e t n Y 1 Y 2 Y k deno te th e n um b e r o f d ist in c t va lu e s in ΗΗΗw h e re ΗΗΗa re (( ) )1988. , T h u s, e st im a to r an d th e ir fu n c t io n s even fo r a sim p le ca se R o b e r t so n W r igh t an d D yk st ra ( )in stead o f seek in g th e b ia s te rm ex ac t ly, w e co n side r th e m e tho d o f e st im a t in g B Η. T h e ca se s w h e re k 1 2 k 1 2 υ υ υ υ υ (, , , . , = 5 = < = < , , , ,co n st ru c ted b y Y 1 Y 2 Y kFo r ex am p lesuppo se th a t k an d ΗΗΗΗΗth en n Y 1 Y 2 1 2 3 4 5 ) = 3.Y 5 :N ow th e fo llow in g re su lt is o b ta in ed ƒTheorem Suppo se th a t Y i is a no rm a lly d ist r ib u ted ran dom va r iab le w ith m ean Ηi an d va r ian ce ?i n i fo r = 1, 2, , . ikT h en () ()B Η= E n Y , Y , , Y .Η1 2 k ( ) , , , , , T h e n um b e r n Y 1 Y 2 Y k is th e d im en sio n o f th e face o f Cw h e re Ηb e lo n g s to an d is an ()() . , :u n b ia sed e st im a to r o f B ΗFo r e st im a t in g B Ηth e fo llow in g e st im a to r m ay b e em p lo yed 1 ) (n Y , Y , , Y .Ν= i i i B k ? 1 2 ki, , in i 1k 7i= 1 () ()= ., , O b v io u slyf rom th e def in it io n E ΗΝB B Η 3 T h e ca se o f th e va r ian ce s w ith com m o n u n k now n p a ram e te r , = 1, , Suppo se th a t in dep en den t ran dom sam p le s x ij j n i f rom a no rm a l d ist r ib u t io n w ith u n k now n 2 2 , = 1, 2, , . , m ean Ηian d a va r ian ce Ρ?i p ro v ided ?i is k now n an d Ρis u n k now n fo r ikS im ila r ly a s th e ca se , o f k now n va r ian ce sth e k tup le s o f th e u n re st r ic ted m ax im um lik e lihoo d e st im a to r an d th e o rde r δ υ , , re st r ic ted m ax im um lik e lihoo d e st im a to rdeno t in g b y Ηan d Ηre sp ec t ive lya re ju st th e sam e w ith th e .ca se o f k now n va r ian ce s 2 ζ , Sp ec if ica llyth e o rde r re st r ic ted m ax im um lik e lihoo d e st im a to r Ρis g iven b y n n k k k iiυ 2 δ 2 δ υ 2 () () ( ) x i j - Ηi x i j - Ηi Ηi - Ηi 1 1 1 2 ζΡ= = + ?? ?? ? N ?N ?N ?iii i= 1 j = 1 i= 1 j = 1 i= 1 T h en th e b ia s te rm is eva lu a ted a s υ υ < Η, Η- Η>1 - 1 - υυ 1 2 2 2 () () (()) B Η, Ρ= E + O N = 1 + E < Η, Η- Η> + O N Η, ΡΗ, Ρζ22 ΡΡ ( ( ) )( ) 1999 . , , 1 + , see A n rak u T h u sa s an app ro x im a te ly u n b ia sed e st im a to r o f B Ηw e m ay u se ΝB .w h e re ΝB is def in ed in th e p rev io u s sec t io n 4 Iden t if ica t io n o f d is t in c t p a ram e te r s k - 1 , 2T o de tec t th e co n f igu ra t io n o f p a ram e te r sw e com p a re th e m o de ls fo r co n f igu ra t io n o f ,p a ram e te r s Η= = Η; Η= = ΗΦ Η; Η= = ΗΦ Η= Η; ; ΗΦ Φ Η. 1 k1 k - 1 k1 k - 2 k - 1 k1 k If th e m o de l Η= = ΗΦ Η= = ΗΦ Φ Η= = Η1 m m + 1 m + m m + + m + 1 k 1 11 2 1h- 1 , , , , .Ηm + 1 Ηm + m + 1 Ηm + + m + 1 w ill b e rega rded a s th e ch an ge po in t s h a s b een se lec tedtho se po in t s a t 1121h - 1 ( ) 1 1987, Exam p le w e app ly th e n ew c r ite r io n to th e da ta in T ab le f rom Yo sh im u ra w h ich show s . th e am o u n t o f h em o g lo b in in b loo d o f an im a ls fo r fo u r do se leve lsT h e da ta w a s a lso an a lyzed in ( )( ) . 1993. . 1988, K ik u ch i e t a lA s w ith R o b e r t so n e t a lw e a ssum e th a t th e da ta w a s o b ta in ed f rom 2 , , , Ε Ε Ε .no rm a l d ist r ib u t io n s w ith an u n k now n comm o n va r ian ce Ρan d m ean s Λ0 Λ1 Λ2 Λ3 w ith Λ0 Λ1 Λ2 Λ3 . = -T h e dec rea se in th e am o u n t o f h em o g lo b in m ay b e rega rded a s th e re su lt o f th e adve r se effec tSe t Ηi ( )= 1, 2, , 4.Λi- 1 i 1 Table Am oun t of Hem og lob in - 1 - 1 () ()ƒ?ƒ?do se d i m gk gH em o g lo b in in b loo dm gk g 0 153 153 152 156 158 151 151 150 148 157 5 158 152 152 152 151 151 157 147 155 146 10 153 146 138 152 140 146 156 142 147 153 20 137 139 141 141 143 133 147 144 151 156 :N ow w e deno te th e po ssib le m o de ls o f co n f igu ra t io n s o f Ηs a s fo llow M ?Η= Η= Η= Η, M ?Η= Η= ΗΦ Η; M ?Η= ΗΦ Η= Η, 0 1 2 3 4 1 1 2 3 42 1 2 3 4 M ?ΗΦ Η= Η= Η, M ?Η= ΗΦ ΗΦ Η; M ?ΗΦ Η= ΗΦ Η, 3 3 1 2 3 4 4 1 2 3 45 1 2 4 Φ ΗΦ Η= Η, M ?ΗΦ ΗΦ ΗΦ Η.M ?Η6 1 2 3 4 7 1 2 3 4 , , , - 152. 90, - 152. 10, - 147. 30 - 143. 20T h e sam p le m ean s f rom Η1 Η2 Η3 Η4 a re g iven b y an d () - - 1 ,, ,. , :re sp ec t ive lyT h e va lu e s o f o u r c r ite r io n L ΗΝB f rom m o de ls M 0 M 1 M 7 a re g iven a s fo llow s () () () () ()- 132. 3 M - 126. 8 M , - 125. 2 M , - 130. 3 M , 10 1 2 3 () () () ()- 124. 0 M , - 126. 3 M , - 125. 5 M , - 124. 3 M .4 5 6 7 = Φ Φ . T h e refo re th e m o de lM 4 th a t is th e co n f igu ra t io n Η1 Η2 Η3 Η4 is se lec tedT h is re su lt co in c ide s w ith (). 1993.K ik u ch i e t a l 5 A M o n te C a r lo s tu dy 2 ( ) , , . , T o e st im a te th e b ia s te rm B ΗΡo n e m ay u se th e boo t st rap m e tho d d irec t lyH ow eve rin , . gen e ra ld irec t app lica t io n s o f th e boo t st rap m e tho d w ill y ie ld a la rge va r ian ceSo w e em p lo y th e ( () )1996.va r ian ce redu c t io n m e tho d see Ko n ish i an d K itagaw a 3 , = 1, 2, , , = 1, , , , = 1, 2, , , = 1, , ij Fo r th e o b se rva t io n s x ij i k j n i le t x i k j n i deno te th e 3 2 ζυ. , , , o b se rva t io n s fo r a boo t st rap sam p leM o reo ve rle t Ηi Ρ3deno te th e m ax im um lik e lihoo d e st im a te s 3 .i j b a sed x S 2 () , , , A boo t st rap e st im a teb y m ean s o f th e va r ian ce redu c t io n m e tho do f B ΗΡw ill b e g iven b y n n k k ii 3 3 ζ2 3 ζ2 υυ() ()lo gf x , Η, Ρ- lo gf x i j i, Η, Ρij i 3 ????i= 1 j = 1 i= 1 j = 1 n n k iki ζ2 3 ζ2υυ)() ( i Ρ- lo gf x Ρ+ lo gf x , Η, , Η, 3 i i j i j ????i= 1 j = 1 i= 1 j = 1 n n kk ii 3 υ 2υ3 2 () () x i j - Ηi x i j - Ηi = + - N . ζ ζ22?? ?? 3 2Ρ2Ρi= 1 j = 1 i= 1 j = 1 3 . L e t deno te ΝB th e boo t st rap e st im a te ave rag in g th e abo ve o ve r a n um b e r o f th e boo t st rap ite ra t io n s 1+ A sim u la t io n stu dy w a s ex ecu ted fo r th e com p a r iso n o f th e e st im a ted m ean s an d va r ian ce s o f ΝB 3 an d ΝB w h en Η1 = Η2 = Η3 = Η4 = 0 fo r 10 000 ite ra t io n s, p ro v ided n i = 10, i = 1, 2, 3, 4. Fo r sim p lic ity, w e 3 em p lo yed th e p a ram e t r ic boo t st rap m e tho d. A s seem f rom th e T ab le 2, ΝB is com p le te ly to ΝB a t lea st .w ith re sp ec t to th e m ean squ a red e r ro r 3 = = = = 0 10 000 2 1+ Table Com par ison of e st im a ted m ean s an d var ian ce of ΝB an d Νwhen Η1 Η2 Η3 Η4 f or itera t ion s B 1+ 2. 00 2. 50 2. 50 2. 50 2. 80 2. 81 2. 89 3. 08 ΝB 0. 000 0. 018 0. 016 0. 017 0. 030 0. 031 0. 034 0. 043m se 3 1. 95 2. 53 2. 52 2. 53 2. 90 2. 90 3. 01 3. 25ΝB 0. 187 0. 380 0. 347 0. 369 0. 507 0. 521 0. 556 0. 677 m se : 100.N o teT h e n um b e r o f th e boo t st rap ite ra t io n is , T o ex am in e th e p e rfo rm an ce o f th e o rde r re st r ic ted in fo rm a t io n c r ite r io n w e ex ecu ted a M o n te = 4. , , , C a r lo sim u la t io n fo r a ca se o f k W e gen e ra ted no rm a l ran dom n um b e r s w ith m ean s Η1 Η2 Η3 Η4 an d () 1. , , , :va r ian ce T h e fo llow in g co n f igu ra t io n s o f p a ram e te r s w e re se lec ted fo r Η1 Η2 Η3 Η4 () () () C 1: 0. 0, 0. 0, 0. 0, 0. 0, C 2: 0. 0, 0. 0, 0. 0, 0. 5, C 3: 0. 0, 0. 0, 0. 5, 0. 5, () (): 0. 0, 0. 5, 0. 5, 0. 5, : 0. 5, 0. 5, 0. 5, 0. 5.C 4C 5 3 8 10000 = 10,T ab le show s th e f requ en c ie s se lec ted a s b e st am o n g m o de ls b a sed o n ite ra t io n s fo r n i = 1, 2, 3, 4. 4 i T ab le show s th e com p a r iso n o f f requ en c ie s th a t th e f ir st ch an ge po in t s a re co r rec t ly .se lec ted fo r th e A IC m e tho d an d o u r m e tho d 3 8 10 000 Table Frequen c ie s se lec ted by our cr iter ion a s be st am on g can d ida te s f or itera t ion s Co nf igu ra t io n Se lec ted m o de l M 1 M 2 M 3 M 4 M 5 M 6 M 7 H o f p a ram e te r s 7 150 949 813 960 38 59 31 0 C 1 C 2 2 890 4 323 1 109 733 346 518 69 12 C 3 2 166 1 112 4 307 1 161 484 283 453 34 2 940 735 1 103 4 268 82 509 344 19 C 4 C 5 38 606 1 725 611 1 728 2 292 1 710 1 290 : .N o teT h e bo ld d ig it s in d ica te th e co r rec t f requ en c ie s F rom th e tab le s w e can see th a t o u r m e tho d is sup e r io r to th e A IC m e tho d w h en th e re ex ist s an . , o rde r re st r ic t io nO f co u r seth e a sse r t io n o f th e sup e r io r ity o f an m e tho d to th e o th e r s w ill b e . , .re st r ic t iveH ow eve ro u r m e tho d seem s m o re p refe rab le to de tec t ch an ge po in t Table 4 Com par ison of correc tly se lec ted f requen c ie s f or the f ir st chan ge po in t 10 000 by our cr iter ion an d A IC m e thod over itera t ion s 2A IC O R A IC Η Η1 Η2 Η3 Η4 Η1 Η2 Η3 Η4 408 428 755 8 409 1 050 851 949 7 150 C 1 491 715 4 330 4 464 1 332 1 455 4 323 2 890 C 2 726 3 779 1 590 3 905 1 931 4 791 1 112 2 166 C 3 3 178 1 051 896 4 875 5 140 1 185 735 2 994 C 4 C 5 3 958 4 191 1 645 206 5 930 3 453 606 38 ( ) = 1, 2, 3: N o teT h e bo ld d ig it s in d ica te th e co r rec t f requ en c ie s Ηi ico r re spo n d s to th e f ir st ch an ge , .po in tp ro v ided Η4 co r re spo n d s to no ch an ge po in t Ref eren ce s 1 . [. 2A k a ik e HInfo rm a t io n th eo ry and an ex ten sio n o f th e m ax im um lik e lihoo d p r inc ip e A In nd I n te rna t iona l [, 1973, 267 , 281.S ym p os ium on I nf orm a t ion T h eory C 2 . [. , 1974, 19:A k a ik e HA new L oo k a t th e sta t ist ica l m o de l iden t if ica t io n J I E E E T ransac t ions on A u tom a t ic C on t rol 716, 723. 3 . [. , 1999, 86: 141A n rak u KA n info rm a t io n c r ite r io n fo r p a ram e te r s unde r a sam p le o rde r re st r ic t io n J B iom e t r ik a , 152. 4 , , . 22[.K ik uch i Y Yanagaw a T N ish iyam a HE te rm in ing th e no o b se rvedadve r seeffec t leve l in co n t inuo u s re spo n se A [. 1993, 345, 356.In S ta t is t ica l S c iences and D a ta A na ly s is C 5 , . [. , 1996, 83: 875, 890.Ko n ish i SK itagaw a GGene ra lised info rm a t io n c r ite r ia in m o de l se lec t io n J B iom e t r ik a 6 , . [. , 1951, 22: 79, 86.Ku llback SL e ib le r R AO n info rm a t io n and suff ic iency J A nna ls of M a th em a t ica l S ta t is t ics 7 , , . [. : , 1988.R o be r t so n T W r igh t F T D yk st ra R LO rd e r R es t r ic ted S ta t is t ica l I nf e rence M N ew Yo rkW iley 8 , , . [. : , 1986.Sak am o to Y Ish igu ro M K itagaw a GA k a ik e I nf orm a t ion C r ite r ion S ta t is t ics M T o k yoD R e ide l 9 , . [ . T ak euch iKD ist r ibu t io n o f info rm a t io n num be r sta t ist ic s and c r ite r ia fo r adequacy o f m o de ls J M a th em a t ica l , 1976, 153: 12, 18.S c iences 10 , , . 222[ . Yanagaw a T K ik uch i Y B row n K GN o o b se rvedadve r seeffec t leve ls in seve r ity da ta J J ou rna l of th e , 1997, 92: 449, 454.A m e r ican S ta t is t ica l A ssoc ia t ion 11 . [. : 2, 1987.Yo sh im u ra IS ta t is t ica l A na ly s is of T ox icology and D ru g Ef f ec t M T o k yoSc ien t istsya 12 , . 222Yanagaw a T K ik uch i YS ta t ist ica l issue s o n th e de te rm ina t io n o f th e no o b se rvedadve r seeffec t leve ls in to x ico lo gy [. , 2001, 319, 325.J E nv ironm e t r ics 有序参数的一类 型信息准则A IC 陈 冬 ( )北京联合大学 基础部, 北京 100101 () : 给出一种基于信息量准则 原理的参数估计方法. 该方法给出了具有有序分布参数的正 摘要A k a ik e A IC 态分布的一种偏差修正项的精确无偏估计量; 且兼有自助抽样的功能得到了开发.A IC () 关键词: 信息量准则 ; 自助抽样; 参数估计A k a ik e A IC