ESTIMATION OF ABILITY USING PSEUDOCOUNTS IN ITEM RESPONSE THEORY Title Author(s) Citation Behaviormetrika (2014), 41(1): 131-146 Issue Date URL Rights Ogasawara, Haruhiko 2014-09 http://hdl.handle.net/10252/5380 ファイルの原本はJSTで公開されているものである。 This document is downloaded at: 2015-01-31T23:48:18Z Barrel - Otaru University of Commerce Academic Collections B e h a v i o r m e t r i k a , , , Vo1 .41 No.1 2014 131-146 ESTIMATIONOFABILITYUSINGPSEUDOCOUNTSINITEM RESPONSETHEORY HaruhikoOgasawara* Amethodf o re s t i m a t i o no fa b i l i t yu s i n gp s e u d o c o u n t si nd i c h o t o m o u si t e mr e s p o n s e m o d e l si sg i v e nwhena s s o c i a t e di t e mp a r a m e t e r sa r eknowno re s t i m a t e db yas e p a r a t e c a l i b r a t i o ns a m p l eo fe x a m i n e e sw i t ht h es i z eo fa na p p r o p r i a t eo r d e r .T hep s e u d o c o u n tm i n i m i z i n gt h ea s y m p t o t i cmeans q u a r ee r r o ri sa l g e b r a i c a l l yo b t a i n e d .Though , af i x e dl o w e rboundf o rt h ep s e u d o c o u n t t h ep s e u d o c o u n td e p e n d so nunknowna b i l i t y i sd e r i v e du n d e rt h el o g i s t i cm o d e lw i t he q u i v a l e n ti t e m s . Thel o w e rboundi snum 町田 i c a l l ys h o w nt ob er e a s o n a b l eu n d e rt h e3 p a r a m e t e rl o g i s t i cm o d e lw i t ha n dw i t h o u t m o d e lm i s s p e c i f i c a t i o n . 1 . Introduction I nitemr e s p o n s et h e o r y( I R T )e s t i m a t i o no ft h ea b i l i t yo rp r o f i c i e n c yl e v e lo fan examineei soneo ft h emainp u r p o s e so fthea s s o c i a t e da b i l i t yt e s t . Amongv a r i o u s , t h emaximuml i k e l i h o o de s t i m a t o r(MLE;Lordう 1 9 5 3 )hasbeen e s t i m a t o r so fa b i l i t y ab a s i co n e . Themaximumap o s t e r i o r io rBayesmodale s t i m a t o r(BME;Samejima 1969う Chapter2 ;Bock&A i t k i n, 1 9 8 1 )i sa l s of a m i l i a r, wheret h estandardnormal p r i o ri st y p i c a l l yu s e d .Thes o c a l l e dweightedl i k e l i h o o d(WL)byWarm( 1 9 8 9 )g i v e s t h eW Le s t i m a t o r(WLE)removingt h easymptoticb i a so ft h ec o r r e s p o n d i n gMLE 山 ( 1 9 8 9 )methodcanbes e e na sa whent h eIRTmodeli sc o r r e c t l ys p e c i f i e d . Warrr i r t h1 9 9 3 ) . s p e c i a lc a s eo ft h eweighteds c o r eo rt h ep e n a l i z e dl i k e l i h o o d( s e ee . g F Ogasawara ( 2 0 1 2 a )g a v et h e asymptotic p r o p e r t i e so ft h e MLEunder p o s s i b l e BMEand modelm i s s p e c i f i c a t i o n( p . m . m . ) . Ogasawara( 2 0 1 3 a )d e a l twitht h eMLE, WLEa ss p e c i a lc a s e so ft h ee s t i m a t o rbyt h eweighteds c o r ewithag e n e r a lw e i g h tt h a t c o r r e s p o n d st ot h ef i r s td e r i v a t i v eo ft h el o gp r i o rd e n s i t ywithr e s p e c tt ot h ea b i l i t y 白e df o r m u l a t i o n, Ogasawara( 20 l3 a ) parameteri nt h ec a s eo ft h eBME.Undert h i su n i t h easymptoticmeans q u a r ee r r o r(AMSE)o ft h e g a v et h easymptoticp r o p e r t i e se . g ., e s t i m a t o rwitht h eg e n e r a lw e i g h t . 抗 i temp紅 a r 担 a metersa r eassumed七obek 王n owno rt ohave五xed I nt h eabovemethods, V 刊a l u 悶e b i 出 l 日 i t 匂 ye s t i m a t o rwhent h ea s s o c i a t e ditemparametersa r ee s t i m a t e dbyas e p a r a t e t h e油 a o ritemc a l i b r a t i o n . Ogasawara( 2 0 1 3 b )a l s oshowed sampleo fexamineeswiths i z eN f 2 th 抗 w henN i so fo r d e rO ( η5/ ), whereηist h enumbero fi t e m sぅ t h easymptoticc u う う ・ う う K巴y Wordsα ndPhα s e s :3 p a r a m e t e rl o g i s t i cm o d e l jmeans q u a r ee r r o r ja s y m p t o t i cb i a s ja s y m p t o t i c v a r i a n c e jp s e u d o c o u n t jB a y e sm o d a l ja b i l i t y . *O t a r uU n i v e r s i t yo fC o m m e r c e T h i sw o r kw a sp a r t i a l l ys u p p o r t e db yaG r a n t i n A i df o rS c i e n t i f i cR e s e a r c hf r o mt h eJ a p a n e s eM i n i s 七r y o fE d u c a t i o n, C u l t u r e, S p o r t s, S c i e n c ea n dT e c h n o l o g y , N o . 2 3 5 0 0 3 41 . A u t h o r ' sa d d r 白日 s :D e p a r t m e n to fI n f o r m a t i o n乱n dM a n a g e m e n tS c i e n c巴, O t a r uU n i v e r s i t yo fComm巴r c e, 3 5 2 1, M i d o r i, O t a r u0 4 7 8 5 0 1, J a p a n .E m a i l :h o g a s a @ r e s . o t a r u u c . a c . j p , " H .O g a s a w a r a 1 3 2 mulantsupt ot h ef o u r t ho r d e randt h eh i g h e r o r d e ra s y m p t o t i cv a r i a n c eo ft h ea b i l i t y e s t i m a t o rwitht h eg e n e r a lw e i g h tu s i n ge s t i m a t e ditemp a r a m e t e r sa r ei d e n t i c a lt o t h o s ewhent h eitemp a r a m e t e r sa r elmown. h edichotomousr e s p o n s emodeli sused, wheret h e2xnc o n t i n g e n c y I nt h i spaper t t a b l ef o rc o r r e c tandi n c o r r e c tr e s p o n s e sbyanexamineei sf o r m u l a t e dwithni t e m s b e i n gg e n e r a l l yd i s t i n c t .I nt h et a b l e, t h e r ea r en1' sandn0s .The2x1c o l l a p s e d wheret h e t a b l ef o rt h enumberso fc o r r e c tandi n c o r r e c tr e s p o n s e scanbeo b t a i n e d, h en u m b e r c o r r e c ts c o r ei sn o tn e c e s s a r i l yt h e sumo ft h etwof r e q u e n c i e si s叫 thought 伍c i e n ts t a t i s t i cf o rt h ea b i l i t yparameteru n l e s ss p e c i a lmodelse . g , ・t h e1 p a r a m e t e r s u l o g i s t i cmodel(1PLM)a r eu s e d . Asas p e c i a lc a s eo ft h e1PLM, t h el o g i s t i cmodel 市 a l e n ti t e m sw i t h o u tg u e s s i n gparameters仏 ( LME;Bi 註 r 口由 I a u r 立 r I 叫 withe q l t ω o t 吐 h 珂eb i n o m i a le r r o rmodel( L o r d& No v i c l 王 ぅ 1 9 6 8ぅ Chapter2 3 ) . A pseudocount(PC)i sana r t i f i c i a lf r e q u e n c yt obeaddedt ot h ec e l l so ft h ec o n b e t t e r "e s t i m a t i o no fa s s o c i a t e dp a r a m e t e r s .I nt h e t i n g e n c yt a b l ei no r d e rt ohave“ v a r i o u sv a l u e so ft h e c a s eo ft h eb i n o m i a lp r o p o r t i o nwi tht h e2x1c o n t i n g e n c yt a b l e, pseudocounthavebeenused, whichw i l lbea d d r ・ e s s e dl a t e r .Thep seudocountmethod cana l s obes e e na saw e i g h t e ds c o r emethod, whichg i v e st h ea s y m p t o t i cp r o p e r t i e s o ft h ea b i l i t ye s t i m a t o rwitht h ep s e u d o c o u n t . h epseudocountwithminimizedAMSEi sd e r i v e d .U n f o r t u n a t e l y , I nt h i spaperぅ t t h eo p t i m a lpseudocountdependsont h ep o p u l a t i o na b i l i t yt obee s t i m a t e d .However, whent h eLMEh o l d s, a五xedl o w e rboundo ft h epseudocounti sf o u n d . Thel o w e r boundando t h e rv a l u e so f五xedpseudocountsa r ei l l u s t r a t e di nt h ec a s eo ft h ef a m i l i a r 3 p a r a m e t e rl o g i s t i cmodel(3PLM)withandw i t h o u tm.m. wheret h eitemparame t e r sareassumedt obeknowno re s t i m a t e dundert h ec o n d i t i o no fN = O ( η 5 / 2 ) . Asymptoticands i m u l a t e de s t i m a t i o ne r r o r si nan u m e r i c a li l l u s t r a t i o nshowt h a tt h e l o w e rboundo b t a i n e dundert h eLMEi sr e a s o n a b l eeveni nt h ec a s eo ft h e3PLMwith andw i t h o u tm.m. う う う 2 . Estimationusingpseudocounts L e tUm(m= 1,・・・ぅ η )bet h edichotomousv a r i a b l et a k i n gv a l u e so f1and0f o r c o r r e c tandi n c o r r e c tr e s p o n s e s, r e s p e c t i v e l yt ot h em-thitembyanexamineewith a b i l i t y8 .I nt h ec a s eo ft h e3PLM )=Cm十イ Pm三 P r ( U 1 8αm, b r Cm m=1 ぅ 川 1C m /n Qm三 1-Pm (m= 1ぅ・・・ ,n) ( 2 . 1 ) whereD =1 .7 ;andαm, b ndCm a r et h eitemp a r a m e t e r sf o rt h em-thi t e m . The ma う mainr e s u l t si nt h i spapera l s oh o l dwhent h eitemp a r a m e t e r sa r er e p l a c e dbyt h esam( η 5 / 2 )(Ogasawara, 2 0 1 3 b ) .Then o t a t i o n p l ec o u n t e r p a r t sundert h ec o n d i t i o nN = O e . g .う 五xedαmr a t h e rthane s t i m a t e dam i sf o rs i m p l i c i t y .Undermodelm i s s p e c i f i c a t i o n ( m . m . ), ana l t e r n a t i v ep r o b a b i l i t yP tm f o rc o r r e c tr e s p o n s ei sd e f i n e d部 PT ぅ QTm三 1 -PTm (m= 1,・・・ぅ η)ぅ m三 ET(U m) ( 2 . 2 ) ESTIMATIONOFABILITYUSINGPSEUDOCOUNTSI NITEMRESPONSETHEORY 1 3 3 whereE T ( ' )i st h ee x p e c t a t i o nu s i n gt h et r u ed i s t r i b u t i o no fUma teandPTmチPm f o ra tl e a s tanitem( E ( . )willbeusedundercorrectmodelspeci五cation(c.m.s.)). De 五ne♂ a st h et o t a lnumbero ft h epseudocountsf o rt h e2x1c o 1 1 a p s e dc o n t i n *i sn o tn e c e s s a r i l yi n t e g e r v a l u e d . Thati s, e a c hc e 1 1has0 . 5 c * gencyt a b l eぅ wherec f o ri t spseudocoUI 止 I nt h ec a s eo ft h eb i n o m i a lpropo 凶 o n, ♂ = 1( 0 . 5f o re a c h 19 5 6 )andAnscombe( 1 9 5 6 )s u c ht h a tt h ea s y m p t o t i c c e 1 1 )wasd e r i v e dbyHaldane( b i a so fo r d e r0(n-1)f o rt h esamplel o g i tv a n i s h e s . Thev a l u ec *= 0 . 5( 0 . 2 5f o re a c h c e 1 1 )i susedbyH i t c h c o c k( 1 9 6 2 )i nt h ec a s eo fl o g i s t i cr e g r e s s i o n .A g r e s t iandC o u 1 1 ( 1 9 9 8 )( s e ea l s oAg 1 'e 凶 &C a f f o, 2 0 0 0 )proposed♂ =4( 2f o1' e a c hc e 1 1 )f o ri n t e1'v a l ぅ : yt h ev a l u e0 . 1o ft h epseudocountf o r e s t i m a t i o no ft h eb i n o m i a lp r o p o r t i o n .R e c e n t l c e( 2 0 0 7 ) . e a c hc e 1 1o ft h e2x2t a b l ewasuse dbyBonettandP1'i Fo1' t h e3PLMwithg e n e r a 1 1 yd i s t i n c ti t e m st h et o t a lnumbe1' o fp s e u d o c o u n t si s もob ee q u a 1 1 yd i s t r i b u t e dt on i t e m sa si nt h eabovec a s e s .Thati s, i nt h e2x' nt a b l e, 1 e a c hc e 1 1h a sn-0 . 5 c *f o1' i t sp s e u d o c o u n t .L e tLpcbet h e( p s e u d o )l i k e l i h o o do fe whent h epseudocounti su s e d .Then, Lpci sf o r m a 1 1 yw1'i t t e na s う 2 1111/111 nJ り 5 n n u ハw pm n H M ,f / EBBEES 山 /lE ノ 、111 、、、‘ V トm 山 ハJFm n //1111¥ 日 目 vb 一 一 n u 十 トm 九 ハw nu 目 十 7fAσ JrL nH何日 一一一一一 p c L 二 日 whe1'eL= = 1 P J 1 7 7 3 Q U U mi st h e田 工 叫 l i k e l i h o o d01' Lpcwithc *= 0 ;f ( e )= 0(1) i ss e e na s ap r i o rd e n s i t yo ri t sp r o p o r t i o n a l from B a y e s i a np o i n to fv i e w ; and f m ( e )三 (P Q m ) O . 5 c * i s a l s o s e e n a s a n i n d e p e n d e n t p r i o r f o r t h e m t h i t e m . Note m t h a tf ( e )i st h eg e o m e t r i cmeano ff m ( e )(m= 1ぅ・・・ぅ T仏 andbecomese q u a lt of m( e ) wheni t e m sa r ee q u i v a l e n te . g ,・ under吐1 eLME. 凶l y ぅf m ( e )becomest y p i c a lp r i o r si nsomec a s e s .Whent h e2 p a r a m e t e rl o g i s Actl t i cmodel(2PLM)h o l d s, D 2 α ~PmQm i st h eF i s h e ro ritemi n f o r m a t i o nf o rt h em自由 i t e m . Then, ( P m Q m ) O . 5withc *= 1i sp r o p o r t i o n a lt ot h eJ e f f r e y s( 1 9 4 6 ;1 9 6 1ぅ S e c t i o n3 . 1 0 )n o n i n f o r m a t i v ep r i o rf o rt h em-thi t e m . Thati sぅ ♂ =1i n( 2 . 3 )g i v e st h e g e o m e t r i cmeano ft h ei n d e p e n d e n tJ e f f r e y sp r i o r swhereast h eu s u a lJ e f f r e y sp r i o ri s h esumo ft h eηitemi n f o r m a t i o n sa v e r a g e do v e r g i v e nfromt h et e s ti n f o r m a t i o ni .e .t i t e m so rt h ea r i t h m e t i cmeano ft h eitemi n f o r m a t i o n s : う LPm'2/( Z三 η 1 ( 2. 4 ) whereP m '= θPm/δe .Underthe2PLMう ( 2. 4 )becomes I=n- 1 D 2 乞 α ~PmQm ( 2 . 5 ) I ti s known t h a tt h eJ e f f r e y s Bayes modal e s t i m a t o r (JME) under c a n o n i c a l p a r a m e t r i z a t i o ni nt h ee x p o n e n t i a lf a m i l yhasz e r oa s y m p t o t i cb i a so fo r d e rO(η1) 1 9 9 3 ;i nt h ec u r r e n tpaper, t h eBMEr e f e r st oo n l yt h e whent h emodelh o l d s( F i r t h, H .O g a s a w a r a 1 3 4 e s t i m a t o ru s i n gt h es t a n d a r dnormalp r i o r ) .Thea b i l i t ye s t i m a t o rwithf i x e ditemp a t h ec o r r e s p o n d i n ge s t i m a t o r r a m e t e r si nt h e2PLMb e l o n g st ot h i sf a m i l y . However, WarmsWLEundert h e2PLMbecomes i nt h e3PLMd o e sn o tb e l o n gt oi t . Thati s, i d e n t i c a lt ot h eJM E .S i n c eundert h eLMEぅ f ( 8 )with♂ =1becomesproportionalt o i a sr e d u c 七i o nf o rt h ea b i l i t ye s t i m a t o re v e nundert h e3PLM t h eu s u a lJe f f r e y sp r i o rぅ b 宜 ' e r e n ti t e m si se x p e c t e dt osomee x t e n twhenc * 1i su s e d, whichw i l lbe withd i r . n u m e r i c a l l yi l l u s t r a t e dl a t e I ti sknownt h a tt h eb i a so ft h eMLEo fa b i l i t yi sp o s i t i v e l yc o r r e l a t e dwitht h e p o p u l a t i o na b i l i t y8 L o r d, 1 9 8 3 ; Ogasawara, 2 0 1 2 a ) . Thati sぅ whenD oi sp o s i t i v e 0( t h eMLEt e n d st obel a r g e r( s m a l l e r )thanD o・ Equalp seudocountsf o r ( n e g a t i v e ), t h etwoc e l l so fc o r r e c tandi n c o r r e c tr e s p o n s e shavet h ee f f e c to fc e n t e r i n gt h eMLE t o w a r d sO . Whenc * 2,f m ( D ) PmQmwhichi sp r o p o r t i o n a lt ot h ei n f o r m a t i v el o g i s t i c whosec u m u l a t i v ed i s t r i b u t i o nf u n c t i o ni sPmundert h e2PLM.I t p r i o rDαmPmQm, ithD = 1 .7, I P m ( D )-φ ( D ) I <0 . 0 1 i sknownt h a twhenαm=lぅ b m = 0w m= 0andc v i c l 王 ぅ 1 9 6 8ぅ I n e q l 凶i 七y ( 1 7 . 2 . 2 ) )ぅ whereφ ( )i st h ec u m u l a t i v e f o ra l lD ( L o r d& No ( D )withc *= 2i sreducedt o d i s t r i b u t i o nf u n c t i o no ft h estandardnorma l . Thus f anapproximateg e o m e t r i cmeano fv a r i o u sindependentnormalp r i o r s .I nt h ec a s eo f t h eb i n o m i a lp r o p o r t i o n,♂=2c o r r e s p o n d st ot h ef i a tb e t ap r i o r wheret h ep o s t e ) / ( η十2 )withp b e i n gt h e r i o rmeano ft h eb i n o m i a lp r o p o r t i o ni sg i v e nby(叩+1 u n b i a s e dsamplep r o p o r t i o n, whichi sa l s od e r i v e dbyL a p l a c e ' sr u l eo fs u c c e s s i o n( s e e e . g ,・ Wilson,1927ぅ p.210). f m ( 8 )ヲswithotherc * ' sa r es e e na sp r i o r so rp e n a l t yf u n c t i o n s I ns i m i l a rmanners, f o rs h r i n k a g ee s t i m a t o r s( s e ee . g・う Gruberぅ 1998,Chapter1 ;Lehmann& C加 e l l a,1 9 9 8, Chapter5 S e c t i o n5 ) whosee f f e c ti ss t r o 時 e ra sc *becomesl a r g e r . De 五nelpca s う う う う う らc 三 η110gLpc= n-11ogL+η110gf ( 8 ) 叫 三 J ι +山 三 l+η 10.5c ♂ 埼* l o g己 人 ( 2 . 6 ) k ぽ 刷 * つl O う st h eg e o m e t r i cmeano fPmQm(m= 1ぅ・・・ ? η )mentionede a rl i e r whereh= 0(1)i (compareio f( 2 . 5 ) undert h e2PLM). Thee s t i m a t o rD s i n gt h ep s e 凶 o count pc u rlpc, whichi so b t a i n e dbyt h es o l u t i o no ft h e (PCE)i sg i v e nbymaximizingLpco f o l l o w i n ge q u a t i o n : βl pρ θ h ' ーニニェ一一一 +η 0 . 5 c *. : θDpc θDpc ゅ 一上 会式{九一日 =η-1 1 0 . 5 c *ー)}凡, =0, 仰 ) whereh '= θh/θD 2 . 7 ), i ti ss e e nt h a tt h enormale q u a t i o nf o rt h eMLE pc.From( sm o d i f i e df o rD e p l a c i n gt h ea c t u a lo b s e r v a t i o nUm with denotedbyDML i pc byr ESTIMATIONOFABILITYUSINGPSEUDOCOUNTSI NITEMRESPONSETHEORY 1 3 5 Um+n-10 . 5 ♂( 1-2P rbyr e p l a c i n gPmi n{ ・ }withPm-n-10 . 5♂ (1-2 P m ) .Unm)o d e rm.m. withPTm(m= 1 , ・ ・ ・ 川 ) , thepopulation80 i sd e f i n e da st h es o l u t i o no f (l /δ8pc)= 0withoutj ( 8 ) . ETθ 3 . Pseudocountsreducingtheasymptoticmeansquareerror . 5 c 叩 / hi sas p e c i a lc a s eo ft h eg e n e r a lw e i g h tg ( 8 )f o r I n( 2 . 7 )i ti sfoundt h a tO pc t h ew e i g h t e ds c o r ed e a l twithbyOgasawara( 2 0 1 3 a )when8= 8 . Thew e i g h t sf o r BME, JME WLEandPCEa r esummarized邸 t h eMLE, う g ( 8 )= 0 f o rML g ( 8 )= -8 f o rB M, g ( 8 )=1 // ( 2 I ) f o rJM, g ( 8 )=j / ( 2 i ) f o rW Landg ( 8 )= c * h ' / ( 2 h ) f o rPC, ぅ ( 3 . 1 ) where 2P θ Z 一一 1十 ( 3i ~ m'Pm" ( 1-2Pm)Pm' . . 一 一 一 U M - K 1 1凡 Q m ( P m Q m ) 2 y J ヲ 日 __-1 ~ P m 'Pm" -jt1 凡 Qm ( 3 . 2 ) β2p~ andP〓 一 一 子 工 δ8'2 Denotet h em-thcumulanto ft h eg e n e r i ce s t i m a t o r8GWu s i n gt h eg e n e r a lw e i g h t g ( 8 )byκm( 8GW) .Then, t h e i ra s y m p t o t i ccumulantsupt ot h es e c o n do r d e ra r eg i v e n a s κl ( e α1十 O(η-2), GW-80)= n-1 ( 3 . 3 ) κ2 ( eGw)= η1α2+η-2α ム2+ O(η-3)(α2=αML2)ぅ 川 whereη一1 α 1i st h ea s y m p t o t i cb i a so fo r d e r0( n-1);η-1α2i st h eu s u a la s y m p t o t i c 1 2 η 1 ぅ ) whichi se q l 凶 t on-αML2byML;andn-αム2i st h eadded v a r i a n c eo fo r d e r0( h i g h e r o r d e ra s y m p t o t i cv a r i a n c eo f8GW, r e s p e c t i v e l y .Thea c t u a le x p r e s s i o n so ft h e Equation( 3. 4 ) , a s y m p t o t i ccumulantsunderp.m.m.a r eg i v e nbyOgasawara( 2 0 1 3 a, AppendixA . 2 ) t h eMSEo f8GWi sg i v e nby From( 3 . 3 ), ET{(eGW-8 2 }= {κ1(OGW-Oo)}2十 κ2 ( e 0) Gw) ( 3. 4 ) =η1αML2十 η-2(αi+αム2 )+ O(η3) 三 MSEO(n-2)( e Gw)+ O(η3)ぅ whereMSE η 2 ) ( ' )i st h eAMSEupt oo r d e rO ( η 2 ) . Thea c t u a le x p r e s s i o no f o( MSEO(n-2)(8Gw)underp.m.m.i sg i v e nbyOgasawara( 20 l3a, Equation( 3 . 1 3 ) )a s 口 2 )川 MSE o( [ … 二 η川 L 2+η2 +αLL1 (θJθ2[¥ 1 ¥ 明 18=8~8; 1 日) (- '>- 2 . ¥l g ' ( 8 ML2-2 g ( 80) 入 句T 0)α T ( 手18=8 一円 0) 0 ( 3 . 5 ) H .O g a s a w a r a 1 3 6 where αMLム2and αML1 a r ed e f i n e ds i m i l a r l yt o αML2; 入 = ET( θ 2 [ / θ 821ニ 1 1 0 ) ; g ' ( 8 )= θg ( 8 ) /θ 8 1向 。 0 Underc .且 S ぅ ・ u s i n gαML2=Z u1 , whereZ oi sZg i v e nby8 . 1 0 / ( 2 z O )( L o r dぅ 0ぅ αML1=1 9 8 3 ;Warm, 1 9 8 9 )andt h eB a r t l e t ti d e n t i t y ぅ ( 3 . 5 )becomes MSEo(n-2) ( eGw)= η一1 7 6 1十η2[αMLム2十α弘 L 1十2 z u 2 g ' ( 8 0 ) 3 2 z _ZU (2 . o ) g ( 8 ug ( 80 ? L 0 '+1 0)+z ( 3 . 6 ) wherez o 'and] 0a r ed e f i n e ds i m i l a r l yt oZ o・Notet h a tn-1α2+η-2(αMLム2十 αむ L 1 )i n 1 1 η n( 3 . 6 )a r eMSEo(n-2 )( e M L ) ' S, thoughαML ム2 ( 3 . 5 )andn- zü 十 2(αMLム2 十 α~L1) i e s p e c t i v e l y . andα1i n( 3 . 5 )a r eg e n e r a l l yd i f f e r e n tfromt h o s ei n( 3 . 6 )r Wheng ( 80)= c * h o '/ ( 2 h ) , whereh o 'andh oa r ep o p u l a t i o nc o u n t e r p a r t sぱ 0fh '仙an o う 九 e s p e c t i v e l y , ( 3 . 6 )underc . m . s .i sw r i t t e na s hr う MSEO(n-2)( e pc)= MSEO(n-2)( e ML) f ぺ 式 . . : 2 θ f d L11 L2 3 z 2 布 房 副 f)=f)o- Z U (2 0 '+] 0 ) jH-2C4262ziz' 十 η2 ( 3 . 7 ) whichi sminimizedwhenc *i s 一車/五三 日 ¥ ι豆 旦( z n '+]0) c : ' : _ ・ h O ' 2¥h o mm h O) ,i o h o '¥ " U , 2) . 目 ・ ( 3 . 8 ) ? 1 4 什す j hnh円" 2 h n = -2 右子十 2十万 (~. I ' n¥ whereh o 'チo i sassumedandi o>0duet oa n o t h e rassumptiono f五n i t e8 0・ I n( 3 . 7 )and( 3 . 8 ), 2 g ( 8 ) h o ' ~~-1 十 1- 2P mn , 0 c * h。 一 山 白 凡 Qm m ( 3 . 9 ) .L ( s e e( 2 . 7 ) )and 2 g ' ( 80 ) θh O ' 1 f ニ 1 τ石 山 δ一一 1÷1-2Rnn/l つ * 弱E J│0=Oozr f ) = f )o I 一一J 十 f 一2 仇九 P 凡 ? 灯 r m J 7 η J f Z J f 斗 2+ ( ο 1一 -2P, 削 川)~凡町 m mJfH(ρ1 一 -2P灯~r )2 一 … tF 自二コ~ l PmQr ( 3 . 1 0 ) . 一 唱 Then, ( 3 . 8 )i sr e w r i t t e na s 合 in 十名古凡' f 一 生 ( 〉 吋 η (合云許可 や+ き ) η r 2U2 fcJ2Mff一 (1;zr12) 0 ' ( 3 . 1 1 ) ESTIMATIONOFABILITYUSINGPSEUDOCOUNTSI NITEMRESPONSETHEORY 1 3 7 + Undert h e3PLMぅ l e t申m三 1 / [ 1 exp{-Dαm(8-b } ] . Thenぅ i n( 3 . 1 1 ), m) Pm'=Dαm( 1 一 cm) 宙m( 1-¥ [ 1m ) う 2 Pmf/ 二 D α ~(1 一 cm )(l ( 3 . 1 2 ) -2雪m)守m( 1一世m ) ' ; 巾 i n c l u d e su此 nown8 sn o to b t a i n e di np r a c t i c e .HowI n( 3 . 8 )o r( 3 . 1 1 )ぅ C 0andi undert h eLME, where e v e r, Pm= P = 1 1+exp{-D α(8-b)}' Q =1-P (m=17・ ・ ・ ぅn )ぅ 一 ¥ / 702ho=PG/(PQ)=D2dPQ?(3.13) i o '=h o '=j o withP'de 五ned邸 Pmf?a五xedl o w e rboundf o rCU : i ri sfounda sf o l l o w s . . Undert h eLMEo f( 3 . 1 3 )witht h ea s s u m p t i o n so f,10 '=1 ] 。 チ Theorem1 f i n i t e8 al o w e rboundo fc U : i ri s3 . 0, f .Using( 3 . 1 3 ), c U : i ni n( 3 . 8 )becomes Proo t o t o H I : ' n i n -2士 一 十5 ' 2 , / '0 oand ( 3 . 1 4 ) う 4 4 '= D3 a3 "=θ2 ,1/δ82 8 1 ニ 8= D a ( 1-6P+ 6P2)PQ. where,10 ( 1-2P)PQand,10 0 Thenぅ ( 3 . 1 4 )i s PQ){D4a4( ( D2 a2 1-6P+6P2)PQ} , " ,, 1-6P+6P2 c L 1 i I 1 = - 2 + 5 = 2 + 5 {D3a3(1-2P)PQP - (1-2P)2 ( 1-2P) 2-2P(1-P) ," " . 4PQ 2 十 5= 3+(1 --"-;¥2 >3 . Q .E .D. ( 1-2P) ( 3 . 1 5 ) I nTheorem1, t h el o w e rbound3c o r r e s p o n d i n gt o8 sn o ta t t a i n e dby 0 土∞ i nt h el a s ti n e q l 凶i t yi n( 3 . 1 5 )becomesi n f i n i t e l y f i n i t e8 0・ Theterm4PQ/(1-2P)2i a t 8 = O . I t l a r g ea s8 a p p r o a c h e s O . T h i s c o r r e s p o n d s t o t h e z e r o b i a s o f 8 ML 0 0 i so fi n t e r e s t七o五ndt h a tt h el o w e rbound3i st h emidpointo ft h ep s e u d o c o u n t sby u l eandA g r e s t iandC o u l l( 1 9 9 8 ) . L a p l a c eうsr h eLMEi sseldomu s e d . I nc u r r e n ta b i l i t yt e s t sf o ra c h i e v e m e n t sbasedonIRT t t h ef a m i l i a r3PLMu s e df r e q u e n t l yi np r a c t i c ei ss i m i l a rt ot h eLMEi na However, 七h el o w e rboundi nTheorem1i se x p e c t e dt obeo fp r a c t i c a lu s e c r u d es e n s e . Thusぅ sn u m e r i c a l l yi l l u s 七r a t e dwithand t osomee x t e n ti nt h ec a s eo ft h e3PLM whichi w i t h o u tm.m.i nt h en e x ts e c t i o n . 二 一 九 う う 4 . Numericalillustrationunderthe3・-parameterl o g i s t i cmodel 五c i a litemp a r a m e t e r so ft h e3PLM, which Numericali l l u s t r a t i o ni sg i v e nu s i n ga r t i 吋 O宜 m a r er a p r a c t i c e( L o r d,1 9 7 5ぅF i g u r e s1 4 ;Kolen& Brennan,2004,T a b l e6 . 5 ) .Thep r o b a b i l i ・ ・ ・9 η )underm.m. a r eg i v e nbyp e r t u r b i 時 t h eterm-Dαm(8-b t i e sPTm(m= 1, m) 均 ム 1 3 8 H.Ogasawara T a b l e1 :V a l u e so fC ; ' ; 1 inm i n i m i z i n gt h ea s y m p t o t i cmeans q u a r ee r r o rf o rt h e3PLM 。 O 1 2 2 -1 -3 3 n 9 . 6 8 1 3 . 9 1 1 . 6 9 2 0 4 0 . 8 1 1 4 7 . 9 0 6. 4 6 3 . 3 5 2 .7 8 1 9. 4 8 1 8 . 0 9 9 6 . 2 7 4 . 0 7 2 . 6 5 2 . 6 0 5 0 41 2 3 . 5 2 2 5 . 6 3 3 . 0 7 2 . 8 3 3 8 . 4 8 6 8 . 4 0 3 . 9 8 1 0 0 T a b l e2 :Asymptoticands i m u l a t e de r r o r si ne s t i m a t i o no fa b i l i t ywhent h e3PLM1 叫 出 (η=2 0 ) (Numbero f ML B M W L PC(l) PC(2) d e l e t e dc a s e s ) ( 1 1 ) ; ' ; 1 in= 6 . 4 6 1, C ml ASE ml ml ml . 3 7 7 9 3 7 9 6 . 3 7 8 9 . HASE . 3 9 8 9 . 3 3 9 9 . 3 5 7 8 . 3 7 8 3 . 3 8 0 9 . 3 5 9 2 Sim 吐a t e d( S D ) . 4 0 7 5 . 3 4 0 6 . 3 -5.9 . 5 6 . 5 -10.9 αム2 . 1 . 9 -5.5 S i m u l a t e d 9 . 3 1 0 . 7 。 = α? S i m u l a t e d HRMSE S i m u l a t e d PC(3) PC(4) PC(8) P C ( c ; ' ; 1 in ) ml . 3 3 5 3 . 3 4 0 7 1 2 . 1 -10.7 ml . 3 1 1 3 . 3 2 4 6 -18. 4 -15.0 ml . 1 8 6 3 . 2 7 5 5 -43.2 -26.8 . 1 1 .1 3 . 2 6 . 1 O . 0 . 1 1 .1 2 . 9 5. 4 3 6 1 8 . 3 7 9 6 . 3 7 9 3 . 3 6 1 7 . 3 4 7 1 . 3 9 9 3 . 3 8 1 4 . 3 6 3 1 . . 4 0 8 1 . 3 5 9 9 . 3 7 8 3 . 3 5 1 0 =2, C~in = 3 . 3 5 6 . 3 5 . 1 2 9 . 3 1 7 . 0 . 3 2 8 5 . 3 4 4 0 . 1 . 2 。 . 3 3 5 8 . 3 4 3 7 ml . 2 4 2 3 . 2 9 2 1 -33.6 2 3 . 0 1 8 . 5 1 2 . 1 . 3 2 3 8 . 3 3 9 8 ( 5 1 8 8 ) ml ml ml ml ml ml ASE . 5 0 0 3 ml ml 5 3 3 2 . 5 2 9 2 . 4 2 8 1 . 2 9 4 0 . 2 2 8 9 HASE . 6 1 3 9 . 0 6 2 7 . S i m u l a t e d( S D ) . 2 8 5 2 . 4 0 5 9 . 4 006 . 3 4 1 2 . 3 0 1 0 . 2 7 1 5 . 2 0 2 9 . 2 8 9 6 5 0 1 5 . 3 5 . 6 11 .9 26.8 -65.6 1 0 4 . 3 2 5 0 . 6 9 8 . 6 1 5 9 . 2 -79.2 αム2 S i m u l a t e d 4 . 2 -36.0 -53.6 -63.9 6 6 . 6 . 5 -67.6 3 7 0 . 7 -83.7 O . 0 9 . 3 3 4 7 5 . 5 3 9 9 . 2 4 7 . 1 6 . 8 5 5 . 0 4. αi 2 . 2 3 . 3 1 8 . 0 3 4 6 . 1 S i m u l a t e d 2 . 1 5 0 . 2 8 . 3 61 .0 1 5 5 . 7 HRMSE 3 7 5 9 . 5 3 3 2 . 5 2 9 3 . 4 5 4 4 . 41 5 3 . 4 224 . 7 7 4 8 . 41 2 4 . 6 2 7 5 . 41 2 8 . 41 0 7 . S i m u l a t e d 45 4 7 . 4 018 . 43 1 6 . 4 756 . 6 5 6 0 . 4 462 . 5 0 6 7 . N o t e .ASE=η-1/2α泣2;HASE=(η一1 α lvIL 2 + η 2 αム2 )ゆ ;HRMSE= {n-1αlvIL 2十n-2(αム2十 α i ) } 1 / 2 ;ML:maximuml i k e l i h o o d ;BM:Bayesmodal;WL:w e i g h t e dl i k e l i h o o d ;PC(c*): e s t i mationu s i n gp s e u d o c o u n t♂. Thes i g n“ ml"i n d i c a t e st h a tt h ev a l u ei si d e n t i c a lt ot h a tby ML.Thes i g n“ "i n d i c前 田 t h 凶 t h ev a l u ei si m a g i n a r y . ← 【 withanaddedindependentv a r i a b l ef o l l o w i n gN(O ぅD 2)andkeepingET([) = O . The generateditemparametersared e a l twitha sknowno n e s . Samples i z e sn= 20, 50 and100areu s e d . Table1g i v e sc i nunderthe3PLMf o r = -3, -2, ・ ・ ・ ,3 .When = 0ぅ thevalues a r er e l a t i v e l yl a r g ewhichi sexpectedfrom ( 3 . 1 5 ) . Wef i n dthatc~由 as afunction o f i sr e f l e c t e dN-shapedratherthanunimoda l . Someofthevaluesaresomewhat smallerthanthelowerbound3f o rtheLME.Sincemostofthevaluesarel a r g e rthan 3o rapproximately3 thelowerboundseemstobepromising. i v easymptoticandsimulatede r r o r so festimationぅ whenM L, B M, Tables2to6g n : e う e e ESTIMATIONOFABILITYUSINGPSEUDOCOUNTSI NITEMRESPONSETHEORY 1 3 9 T a b l e3 :Asymptoticands i m u l a t e de r r o r si ne s t i m a t i o no fa b i l i t ywhent h e3PLMd o e sn o th o l d ( n= 2 0 ) (Numbero f ML B M W L PC(1) PC(2) PC(3) PC(4) PC(8) PC(C , ; ' ;i n ) d e l e t e dc a s e s ) Pm)= . 7 6 2 ( 2 ) e= 1, C , ; ' ;i n= 6. 46 , Cor(PTm, ml ml ASE ml ml ml ml . 3 0 5 7 ml ml 3 0 4 2 . 2 8 6 9 . HASE . 3 2 0 5 . 2 7 1 6 . 3 0 5 0 . 2 6 8 6 . 2 4 8 9 . 1 4 5 5 . 1 9 2 1 3 0 5 9 . S i m u l a t e d( S D ) . 3 0 4 5 . 2 8 9 2 . 2 6 2 2 . 3 2 5 9 . 2 7 5 6 . 2 7 4 8 . 2 2 3 3 . 2 3 6 5 4 . 4 3 . 7 7 . 9 8 . 5 1 2 . 6 2 8 . 9 2 2 . 6 一 . 2 一.4 αム2 . 3 . 1 4 S i m u l a t e d 5 . 1 7 . 0 3 . 9 -7.2 9 . 9 -17. 1 5 . 0 . 0 . 2 . 1 6 . 1 1 .2 3 . 3 6 . 3 2 8 . 2 1 7 . 9 α ? . 0 1 .2 2 . 8 4 . 9 S i m u l a t e d . 2 4 1 6 . 1 1 1. . 1 5 . 2 3 0 4 8 . HRMSE . 3 2 0 8 . 2 9 8 4 . 3 0 5 0 . 2 9 2 2 . 2 8 3 4 . 2 7 8 7 . 3 0 2 9 . 2 8 5 8 3 0 4 6 . 3 0 6 7 . 2 9 4 2 . S i m u l a t e d . 3 2 6 3 . 2 9 8 4 . 2 8 7 3 . 2 8 4 8 . 3 0 0 1 . 2 9 0 8 =2, C Pm)= . 2 9 0 ( 8 1 8 5 ) 2 山 =3 . 3 5, Cor(PTm, ml ml ml 皿l ml 皿l ml ml ASE . 4 8 7 8 5 4 2 6 . 5 3 8 8 . HASE . 6 1 9 9 . 1 5 0 5 . 4 4 3 1 . 3 2 0 0 . 0 9 1 9 . 2 6 3 3 3 5 6 7 . 3 0 5 6 . 2 4 3 5 . S i m u l a t e d( S D ) . 4 3 8 7 . 2 5 6 5 . 3 5 9 9 . 2 7 0 0 . 1 8 0 6 . 2 5 9 9 2 . 6 2 8 6 . 1 2 0 . 9 1 6 . 7 5 4 . 2 -91 .8 2 4 2 . 1 6 7 . 5 5 8 . 5 αム2 4 4 4 . 3 5 S i m u l a t e d 1 8 . 2 6 8 . 9 -43. .5 8 2 . 1 6 8 . 2 7 . 8 6 6 . 0 -71 2 . 0 . 1 4 9 . 3 6 . 3 5 7 . 5 1 0 . 1 3 6 . 3 7 8 . 6 4 1 0 . 2 α 1 S i m u l a t e d 5 . 0 6 . 3 2 3 . 0 4 4 . 3 67. 4 1 6 2 . 0 5 2 . 3 . 2 5 6 . 2 5 4 2 6 . 5 3 9 0 . HRMSE . 6 3 2 5 . 4 0 7 8 . 47 0 7 . 4 394 . 4 5 2 8 . 8 1 1 2 . 43 8 9 3 7 8 3 . 3 8 8 6 . . 4 4 5 2 S i m u l a t e d . 4 3 9 3 . 4 5 4 1 . 3 7 6 8 . 42 8 5 . 4 7 7 4 . 6 6 1 5 12 1 N o t e .ASE= n-/ α昨2 ;HASE= (η1αML2十η2αム2 ) 1 / 2 ;HRMSE= {n-αML2十η 2 ( αム2 十 a i ) }中 ;ML:maximuml i k e l i h o o d ;BM:B a y e sm o d a l ;WL:w e i g h t e dl i k e l i h o o d ;PC(C t i * . Thes i g n“ m l "i n d i c a t e st h a tt h ev a l u ei si d e n t i c a lt ot h a tby mationu s i n gp s e u d o c o u n tc n d i c a t e st h a tt h ev a l u ei si m a g i n a r y . ML.Thes i g nり i 。 つ : 国 W L, andPC(♂)i . e ., PCwith♂ a r eused, where♂ =1, 2, 3ぅ 4andc U : i n ' Inaddition, ♂ =5o r8i susedi nTables2, 3and5tohavec * ' sg r e a t e rthanc~巾 when c~巾 is not 0 ) . No七ethattheresultsby s ol a r g e .I ti se a s i l yseenthatM Li se q u i v a l e n ttoPC( : Uin)areforcomparisonぅ wherec: Ui ngivenby8 sa v a i l a b l ei ntheexperimental PC( c 0i t 吋 at 吐 h 削O 仇u ghuna , v 刊 a 厄 叩i l 油 a 剥 b l ei r 叩 d瓜 a standarde r r o r, whichi srobustunderm.m.;HASE= (n-1αML2+ n-2αム2 )ゆ i sthe 2 η )1/2 higher-orderASE;HRMSE={η1αML2十 η2(αム2+αi)}1/2= (HASE 十 2 αi i sthehigher-orderasymptoticrootMSEuptoorderO(η-3/2). Notethattheasympt o t i crootMSEuptoorderO(η-1/2)i si d e n t i c a ltotheASE . ThesimulatedSEdenotedbySDcorrespondingtoASEandHASEi sthesquare 5 root o ftheunbiasedvarianceo f1 0 estimates f o r eache s t i m a t o r . The simulated sn2(SD2- n-1α2 ) ' Thesimulatedαii sポ timesthesquareo fthesimulated αム2i b i a s . ThesimulatedrootMSEi sgivenbythesquareroot o fthesimulatedmean o f( e 2 . When8三一 2( 8三 f o rη =2 0 )and8= 3, m出 l yc a s e so fnonG W- 8 0) convergencei nestimationo c c u r r e d . Thesec a s e swerenotemployed. Inthet a b l e s, o 1 4 0 H .Ogasawara 七 ywhent h e3PLMh o l d 日( n=5 0 ) T a b l e4 :A s y m p t o t i ca n ds i m u l a t e de r r o r si ne s t i m a t i o no fa b i l i (Numbero f d e l e t e dc a s e s ) ( 5 9 1 ) ASE HASE S i m u l a t e d( S D ) αム2 S i m u l a t e d 2 α 1 S i m u l a t e d HRMSE S i m u l a t e d ( 0 ) ASE HASE S i m u l a t e d( S D ) αム2 S i m u l a t e d ML 。 = BM WL P C ( l ) P C ( 2 ) P C ( 3 ) P C ( 4 ) PC(C~in) 8 . 0 9 -1. C~.tin = 1 . 3 6 0 1 ml ml ml ml ml ml ml . 3 9 1 5 . 3 6 4 0 . 3 8 4 7 . 3 0 2 3 . 3 7 7 9 . 3 7 0 9 . 3 6 3 8 . 2 4 2 2 3 8 3 2 . . 3 9 7 1 . 2 9 7 4 . 3 4 6 3 . 3 7 1 8 . 3 6 1 7 . 3 5 2 7 . 2 6 9 7 7 . 1 4 5 . 8 3 5 8 . 9 9 5 . 7 2 . 7 1 9 . 6 6 . 6 1 7 7 . 6 4 4 2 . 9 21 .3 2 . 9 1 3 . 3 1 4 2 . 3 7 0 . 0 1 0 3 . 0 -24. O . 7 2 . 1 2 5 . 3 1 .3 . 0 . 2 1 0 3 . 5 . 4 . 7 2 . 6 2 7 . 2 . 4 1 .7 71 .3 . 0 3 6 4 0 . 3 8 5 1 . . 3 9 2 5 . 3 1 8 7 . 3 7 7 9 . 3 7 1 0 . 3 6 4 4 3 1 6 3 . 3 9 8 4 . 3 4 6 5 . 3 8 3 6 . 3 6 1 9 . 3 1 5 2 . 3 7 1 8 . 3 5 3 6 . 3 1 8 2 =0, c 9 6 . 2 7 E 出 ml ml ml . 2 8 3 3 ml ml ml ml . 2 8 7 2 . 2 6 3 8 . 2 8 4 0 . 2 8 2 8 . 2 7 3 7 . 2 7 8 3 . 2 6 9 1 2 8 3 9 . 2 8 3 0 . . 2 8 7 5 . 2 6 5 0 . 2 7 8 6 . 2 7 4 3 . 2 7 0 1 . 1 1 4 0 1 . . 8 7 . 1 1 3 5 . 5 2 6 . 7 0 . 4 1 9 . 7 -601 .1 一 5 2 5 . 2 6 . 7 1 2 . 6 1 8 . 2 1 6 8 . 2 6 . 0 . 7 一. . 0 . 0 . 0 . 4 . 6 . 2 O 3 0 4 . 0 α i . 0 . 4 S i m u l a t e d . 0 . 0 . 0 . 1 . 2 4 8 . 0 HRMSE . 2 8 7 2 . 2 8 4 0 . 2 8 2 8 . 2 6 3 8 . 2 7 8 4 . 2 7 4 0 . 2 6 9 5 S i m t u a t e d . 2 8 7 5 . 2 6 5 0 . 2 8 3 9 . 2 8 3 0 . 2 7 8 6 . 2 7 4 4 . 2 7 0 4 . 1 7 9 4 1 2 2 / n N o t e .ASE= n- α;jL;EASE=(n-1αML2+ αム2)1/2;HRMSE={η1向 山 十 n-2(αL i ) } 1 / 2 ;ML:maximuml i k e l i h o o d ;BM:B a y e sm o d a l ;WL:w e i g h t e dl i k e 十α l .2 s i n g p s e u d o c o u n t ♂ l i h o o d ;PC(c e s t i m a t i o nu T h e s i g n m l " i n d i c a t e s t h a t t h e “ つ i g n“ v a l u ei si d e n t i c a lt ot h a tbyML.Thes n d i c a t e st h a tt h ev m a g i n a r y . a l u ei si "i 。 = thenumberso e l e t e dc n t i l1 05 regularobservationswereobtaineda fd a s e su r eshown. z e r onumberso fthed e l e t e dc a s e swhene 2 ThenOlト. 1 a r e a l l d u e t o p e r f e c ts c o r e s, = , i n i t eeML i a s e sa wheref su日a v a i l a b l eandt h e s ec r enotusedf o ro t h e re s t i m a t o r s . Tables2 4and5a r eg i v e nunderc . m . s . whileTables3and6underm.m., where x t e n to o r r e l a t i o no i t e m s . T a ble2 t h ee fm.m. i sshownbythec fPTmandPmo v e r g i v e sther e s u l t sunderc ss m a l la 0 . Whene . m . s . withnbeinga theasymps2 = 1, a l u e sa t o t i cv r ereasonablys i m i l a rt ot h e i rcorrespondingsimulatedv a l u e s . TheSDi s othecorrespondingHASEthantheASE.Ther r er e a s o n a b l e c l o s e rt e s u l t sbyW La i nt h a tnotonlytheasymptoticb i a si sz e r obyc o n s t r u c t i o n (denotedby0i nthe a s s o c i a t e dt a t h e rthan. o ro t h e rroundednumbers) buta a b l e sr 0f l s o αム2i ss m a l l e r h a tbyM Lwiths a l u e s . Ther thant i m i l a rsimulatedv e s u l t sbyPC( r es i m i l a r 1 )a t ot h o s ebyW L, whichi sexpectedt osomee x t e n ts i n c etheya r ei d e n t i c a lunderthe LM ti h a tther e s u l t sbyPC( E .I o r so fi n t e r e s tt o五吋 t 3 )usingthelowerbound3f 4 )i ss m a l l e rthant h o s e theLMEa r es i m i l a rt ot h o s ebyBM.TheHRMSEbyPC( 由 s i m i l a r w i t h s i m l 3 )andB MandgreaterthanbyPC( t e dt e n d e n c i e s . byPC( C ; ; : l i n ) う う ESTIMATIONOFABILITYUSINGPSEUDOCOUNTSI NITEMRESPONSETHEORY 1 4 1 T a b l e5 :A s y m p t o t i cands i m u l a t e de r r o r si ne s t i m a t i o no fa b i l i t ywhent h e3PLMh o l d s(η=5 0 ) (Numbero f d e l e t e dc a s e s ) ( 0 ) ASE HASE S i m u l a t e d( S D ) αム2 S i m u l a t e d ML BM W L PC(l) PC(2) PC(3) PC(4) PC(5) P C ( C ; ' ; u n ) =4.07 8= 1 ,c E 山 ml ml ml . 2 7 3 3 ml ml ml ml ml 2 7 4 3 . 2 7 1 7 . . 2 7 7 8 . 2 5 4 5 . 2 6 5 5 . 2 5 9 1 . 2 5 2 5 . 2 4 5 8 . 2 5 2 1 2 7 3 5 . 2 7 1 4 . 2 6 5 3 . 2 5 4 0 . . 2 7 7 9 . 2 5 5 3 . 2 5 9 5 . 2 4 8 7 . 2 5 3 6 1 .4 2 . 2 1 0 . 6 1 9 . 0 2 7. 4 3 5 . 8 2 7 . 9 6 . 2 2 4 . 8 . 3 2 . 7 1 0 . 9 1 8 . 5 6 . 3 2 3 . 9 2 5 . 5 3 2 . 1 2 6 . 0 O . 3 2 . 6 4 . 3 . 3 1 0 . 0 7 . 3 1 2 3 . 7 1 4 . 9 α i . 0 . 2 S i m u l a t e d . 4 4 1 2. 4 2 0 . 0 1 2 . 9 2 . 3 8 . 7 6. 2 7 4 3 . 2 7 1 9 . 2 6 7 4 . HRMSE . 2 7 8 1 . 2 6 2 3 . 2 6 4 6 . 2 6 3 6 . 2 6 4 4 . 2 6 3 6 S i m u l a t e d . 2 7 8 2 . 2 7 3 5 . 2 7 1 5 . 2 6 7 0 . 2 6 3 5 . 2 6 2 0 . 2 6 4 4 . 2 6 4 3 . 2 6 3 6 . 6 5 ( 2 6 ) =2 C~in =2 ml ml ml ASE . 3 2 9 0 ml ml ml ml ml . 3 5 1 5 . 3 3 4 4 . 3 3 2 1 . 3 1 1 6 . 2 6 5 7 . HASE 2 6 1 7 . 2 8 9 6 . 2 3 9 6 . 2 9 7 4 3 3 1 3 . S i m u l a t e d( S D ) . 3 5 9 0 . 2 7 4 0 . 3 3 4 3 . 3 0 9 5 . 2 9 1 4 . 2 7 6 1 . 2 6 2 8 . 2 9 7 3 αム2 8 . 8 5 . 1 2 9 4 . 1 8 . 0 -61 4 .1 1 2 7 . 2 4 9 . 6 3 8 . 1 -99. . 8 3 . 8 31 S i m u l a t e d .6 .3 -58. 4 8 0 . 1 9 8 . 0 4 9 . 7 ' 8 3 . 0 8 51 O . 7 1 4 . 1 4 4 . 6 9 2 . 0 1 3 2 . 1 4 . 3 7 5 6 . 5 6 . 6 α i S i m u l a t e d . 0 . 8 1 4. 4 41 5. 4 6 7. 4 .8 8 0 . 0 1 2 6 . 7 3 0 . 9 3 3 2 5 . HRMSE . 3 5 3 9 . 3 1 4 9 . 3 3 4 4 . 3 2 0 5 . 3 1 8 9 . 3 2 7 8 . 3 4 6 4 . 3 1 8 2 S i m u l a t e d 3 3 1 8 . 3 1 8 6 . . 3 6 2 0 . 3 1 9 4 . 3 3 4 3 . 3 1 8 8 . 3 2 9 0 . 3 4 6 1 . 3 1 7 4 2 αム2 ) 1 / 2 ; HRMSE { n -1αML2+ N o t e . ASEη-1/2αL22;HASE=(η1αML2十 η η2(αL i ) } 1 / 2 ;ML:maximuml i k e l i h o o d ;BM:B a y e sm o d a l ;WL:w e i g h t e dl i k e l i h o o d ; 十α l .2 e s t i m a t i o nu s e u d o c o u n t♂. Thes i g n“ PC(C s i m l "i n d i c a t e st h a tt h ev a l u ei si d e n t i c a l 時 p つby t ot h a t ML. 。 う o rPC( Inthet a b l e, i ti sseenthatαム2andαif c * )aremonotonicallydecreasingand r e s p e c t i v e l yi f♂ withs i m i l a rsimulatedv i n d i n c r e a s i n g, ntermso a l u e s .Howeverぅ wef a l u ea r enotmonotonicwithr thatHRMSEandi t ssimulatedv e s p e c tt o♂. i s c r e p a n c i e sappeare When = 2 somed s p e c i a l l yi nther e s u l t sbyM L, whichi s sa p r i m a r i l yduet othel a r g enumbero fd e l e t e dc a s e s whichi smanya s5%o fthe a s e s .I B Mg i v e sthes m a l l e s tHRMSE r e g u l a rc ti sseeni nTable2thatwhen = 2, 2 )yieldsthesmallestsimulatedRMSE. 40 1 8 . . 3 7 5 9whilePC( e a l i s t i cconditioni Table3g i v e sr e s u l t sunderar e s s nthatthe3PLMi smoreo rl ti sencouragingt m i s s p e c i f i e di np b i l i t yt e s t s .I os e ethatther r e, r a c t i c ef o ra e s u l t sa nTable2 i nacrudes e n s eぅ s i m i l a rt . Noteagainthatwhen = 2, othosei a smany a s e scorrespondingt fther e g u l a rc a s8185c o8%o a s e swerediscardedduet e r f e c t op m a l l e s tHRMSEandsimulatedRMSEa r egivenbyPC( 4 ) s c o r e s . When = 1ぅ thes w h i l e t h e y a r e 刊 PC om.m g i nbyB MandW L, notby r e s p e c t i v e l y ( c ,・ n )owingt : ' ni . when = 2 Tables4and5showther e s u l t swithn= 50underc . m . s . when = -1 0,1 2 . e う e う e e e e う う 1 4 2 H .Ogasawara T a b l e6 :A s y m p t o t i cands i m u l a t e de r r o r si ne s t i m a t i o no fa b i l i t ywhent h e3PLMd o e sn o th o l d 0 ) (η=,5 (Numbero f ML B M W L PC(l) PC(2) PC(3) PC(4) PC(C~in) d e l e t e dc a s e s ) ( 8 3 ) o r ( P T , m - Pm )= . 5 0 1 -1, C~in = 1 8 . 0 9, c ASE . 3 6 0 8 ml ml ml ml ml ml ml HASE . 2 8 1 2 . 3 4 9 0 . 3 7 1 8 . 3 7 9 0 . 3 6 4 4 . 3 5 6 9 . 3 4 9 2 . 2 1 3 0 S i m u l a t e d( S D ) . 3 8 4 2 . 2 9 7 1 . 3 4 3 3 . 3 7 3 7 . 3 6 4 2 . 3 5 5 5 . 3 4 7 4 . 2 6 7 9 2 0 . 9 2 0 . 2 3 3 . 8 -127.7 6 . 6 7 . 0 2 0 . 6 2 1 2 . 0 αム2 S i m u l a t e d 4 3 . 7 1 0 4 . 7 -30.7 2 3 . 8 6 . 3 -9.4 2 3 . 7 1 4 6 . 0 3 3 . 9 . 2 . 2 . 1 1 .3 . 9 2 . 7 1 2 8. 4 α? S i m u l a t e d 1 .7 31 .2 . 7 . 3 . 0 . 8 2 . 6 7 9 . 7 HRMSE 3 0 4 4 . 3 4 9 1 . 3 7 1 9 . . 3 7 9 7 . 3 6 4 4 . 3 5 7 4 . 3 5 0 7 . 3 1 1 0 S i m u l a t e d . 3 8 5 1 . 3 1 7 4 . 3 4 3 7 . 3 7 3 8 . 3 6 4 2 . 3 5 6 0 . 3 4 8 8 . 3 2 1 9 ( 1 3 ) o r ( P T , m - Pm )= . 47 0 2, cLin=2.657 c ASE ml ml ml . 2 9 5 3 ml ml ml ml HASE . 3 1 2 6 . 2 3 1 7 . 2 9 7 2 . 2 9 5 2 . 2 7 6 7 . 2 5 6 9 . 2 3 5 5 . 2 6 3 9 2 9 7 4 . 2 9 5 0 . S i m u l a t e d( S D ) . 3 1 8 1 . 2 4 5 9 . 2 7 6 4 . 2 6 0 8 . 2 4 7 6 . 2 6 5 9 8 3 . 8 2 . 9 2 6 . 5 2 6 . 3 . 4 4 3 . 8 5 2 . 9 7 9 一 . 1 αム2 7 . 0 4 7 . 9 6 4 . 7 -41 .2 S i m u l a t e d 3 5 . 0 6 6 . 7 3 . 2 一.4 2 8 7 . 1 . 2 2 . 6 1 .8 1 8 . 4 5 2 . 3 1 0 3 . 7 3 8 . 6 αi 1 .7 1 7 . 6 4 S i m u l a t e d 3 . 3 7 2 . 9 . 2 4 6 . 8 8 6 . 6 3 5. 2 9 6 4 . HRMSE . 3 1 4 2 . 2 9 7 5 . 2 9 7 3 . 2 8 9 7 . 2 9 4 9 . 3 1 1 3 . 2 9 1 7 S i m u l a t e d 2 9 9 4 . 2 9 7 6 . 2 9 6 1 . 2 8 8 8 . . 3 2 0 1 . 2 9 4 6 . 3 0 9 8 . 2 9 1 3 ;HASE= (η1αML2十 η 2 α ム2 ) 1 / 2 ;HRMSE= { η一1 向山十 N o t e .ASE=η-1/2αば2 n-2(αム2十 α i ) }ゆ ;ML:皿 aximuml i k e l i h o o d ;BM:B a y e sm o d a l ;WL:w e i g h t e dl i k e i g n“ m l "i n d i c a t e st h a tt h e l i h o o d ;P C ( c * ) :e s t i m a t i o nu s i n gp s e u d o c o u n t♂. Thes h a tbyML. v a l u ei si d e n t i c a l七ot 。 = 。 = rPC( C : Uin)' When Thes m a l l e s tHRMSEandsimulatedRMSEa r egivenbyB Mo 8= -1and0, thevalueo f♂ i nPC( c * )correspondingtoB MinTable4seemsto e f i e c t i n gther e l a t i v e l yl a r g ec : Uin' Table6showstheresultsunder beg r e a t e rthan4r m.m. when8= -1and2 , whicha r efoundtobes i m i l a rt othecorrespondingresu 1 ts i nTables4and5 . 5 . Remarks Ther e s u l t si nthenumericali 1 l u s t r a t i o nshowthatthelowerbound3i sreasonable e s p e c i a l l ywhen8 sa sl a r g ea s2i ndatas i m i l a rtothosei nt h i spape r . When80 i s 0i e a s o n a b l e . Int h i scase,l a r g ec *' s s m a l l e rthano requalto0,♂=4maybemorer r e c o m m e n d e d f o r maynot alwaysbe usei np r a c t i c ethoughc i v e sthe 加 C L 巾 : U i ng *a s m a l l e s tHRMSEunderc . m . s .i nthed a t a . Thisi sbecausethel a r g ec l s og i v e s r e l a t i v e l yl a r g e rb i a s e sthanthosebye . g sfoundt og i v e ,・ BM. Inthetables,B Mi c * a s e scomparabler e s u l t sbyPC( smallMSEsandi nma . c * )with = 3and4 可 c ESTIMATIONOFABILITYUSINGPSEUDOCOUNTSI NITEMRESPONSETHEORY 1 4 3 However whenr e l a t i v e l ys m a l lb i a si sd e s i r e d, W LandPC( c * )with♂ =1and2 maybeu s e d . ,i ti sp o s s i b l et ohavel e s sb i a s e dBMEsbyu s i n gmorer e l a t i v e l yvague Probably l .However, t h ep r i o ro ft h eBMEi sani n f o r m a p r i o r si np l a c eo ft h es t a n d a r dnorma t i v eonebasedont h es u b j e c t i v ep r o b a b i l i t yo f80 fromapureB a y e s i a np o i n to fview, whoser e a s o n a b l eonei sn o ta l w a y se a s i l yo b t a i n e d . Thes t a n d a r dnormalp r i o rf o r t h eBMEi sr e a s o n a b l ei nmanyc a s e si nt h a titemp a r a m e t e r sa r ee s t i m a t e dt y p i c a l l y bym a r g i n a lMLassumingt h es t a n d a r dnormalf o rt h ed i s t r i b u t i o no fa b i l i t y . The U : i nf o rt h e a u t h o rh a st r i e dt oo b t a i naf i x e dl o w e rboundf o rt h eBMEs i m i l a rt oc PCE.Sof a r, howeverぅ 七h el o w e rboundi sn o ta v a i l a b l e . Probablymethodsd i 百 ' e r e n t ,・ empiricalBayesshouldbeinvestigated. fromt h a tu s i n gt h eMSEf o r8GWe . g I nt h et a b l e st h es q u a r e da s y m p t o t i cb i a si sshowna sav a l u ei n d e p e n d e n to fn .The c o r r e s p o n d i n gb i a so fo r d e rO ( η 1 )i sg i v e nbye . g・う ( α i ) 1 / 2 / η =( 2 5 . 3 )山 / 5 0土 0 . 1 f o rB Mwhen8= -1andn = 5 0i nT a b l e4withas i m i l a rs i m u l a t e dv a l u e . The v a l u e0 . 1maybea c c e p t e dbymanyr e s e a r c h e r se s p e c i a l l ywhenwec o n s i d e rt h el a r g e 2 土 t h ec o r r e s p o n d i n gs q u a r e dasympr e d u c t i o no ft h ev a r i a n c eby-95.7/50 -0.038( 2 土 0 . 0 1 0 )whichr e d u c e sHASE= 0 . 3 9 1 5byMLt oHASE= t o t i cb i a si s2 5 . 3 / 5 0 1 tsbyc~lin when 0 . 3 0 2 3byB Mwiths i m i l a rs i m u l a t e dv a l u e s .Thec o r r e s p o n d i n gr e s u 8= -1andη =5 0i nT a b l e4a r eα (i )1/2η=(103.5)1/2/50と 0.2andthereduction o ft h eHASE= 0 . 3 9 1 5byMLt oHASE= 0 . 2 4 2 2byPC( c U : i n )withs i m i l a rs i m u l a t e d sc U : i nweren o ta l w a y srecommendede a r l i e rbyt h e v a l u e s . Thought h el a r g ec sa a u t h o rt h ev a l u eo ft h eb i a s0 . 2maybet o l e r a b l ef o rsomer e s e a r c h e r sc o n s i d e r i n gt h e 国i a l l ys m a l lv a r i a n c e . c o r r e s p o n d i n gs u b s t a Themethodo fe s t i m a t i o nu s i n gt h epseudocounti sa l m o s ta ss i m p l ea st h a tby ML( r e c a l l( 2 . 7 ) ) . Ont h eo t h e rhand, M S E O ( n 2 ) ( 8 p c )( s e e( 3 . 7 ) )i sa sc o m p l i c a t e d a st h o s ef o ro t h e re s t i m a t o r swithv a r i o u sg ( 8 ) ' s( s e e( 3 . 6 ) ) .I tcanbeshownt h a tan e s t i m a t o rwitht h esamea s y m p t o t i ccumulantsupt ot h ef o u r t ho r d e randt h esame addedh i g h e r o r d e ra s y m p t o t i cv a r i a n c ea st h o s eo f8 i s g i v e n b y c o r r e c t i o n o f8 GW I v IL a sf o l l o w s : う う う 判 う 。 ( _δIvIL C-GW三 8 -1 2 g )ぅ η1aML29(0ML)三 eI I v IL+ v IL-n ( 5 . 1 ) where&IvIL 2i st h esamplev e r s i o no f αI v IL 2 .At y p i c a lexampleo f-&IvIL2gi s&IvIL 1ぅ whichi st h esamplev e r s i o no f αI v IL 1byML, y i e l d i n gt h ea s y m p t o t i c a l l yb i a s c o r r e c t e d 1 e s t i m a t o re n & I v I L 1 ・ R e c a l l t h a t g ( 8 ) = 3 / ( 2 I ) f o r W L ぅ w h i c h u n d e r c . m . s . g i v e s I v IL 一αI v IL 2 g ( 8 )= 3 / ( 2 I 2=αIvIL 1 ). TheAMSEo f8 n( 5 . 1 )i smoree a s i l yob 切i n e dthant h a to f8Gw. Fors i m cGWi p l i c i t y , wec o n s i d e rt h ec a s eunderc . m . s .i . e ,・ 8ιGW= 8IvIL+η 1,i-lg, where,i i st h e samplev e r s i o no fi .Then う E{(e } cGW-80? “ =v a r ( e )+η-22ηacov(0MLJ19)+{E(dc-GW)Oo}2+0(η-3) I v IL 1 =η-1 % 0 +η 2 [ αI v IL ム2 +2ηacov(8I v IL , I 1 g )十 { αI v IL 1+ 記 長1 9 ( 80) P l+O(η 3 )ぅ ( 5 . 2 ) H .O g a s a w a r a 1 4 4 wherea c o v (・ ぅ ・ )i st h ea s y m p t o t i cc o v a r i a n c eo fo r d e rO ( η一1 )f o rt h etwov a r i a b l e s . Using : ; ' 19 I u 3 I O ' g ( 8 0 )+I ぷ2 g ' ( 80 ) , ηacov(e )= ( 5 . 3 ) ML, ( 5 . 2 )becomes む 1 η-Qu +η2{αMLム2十 α L1十 2 I u 2 g ' ( 8 0 )-Iu3( 2 九+30)g(80)+ I u2 g ( 8 ) 2 } 0 十O (η-3) ( 5 . 4 ) =M8EO(n-2)( e ( η 3 ) . Gw)十 O Thee s t i m a t o r8 n( 5 . 1 )wasi n t r o d u c e dt ohavesomei n s i g h ti nt h ei n v o l v e d cGWi e x p r e s s i o no f( 3 . 6 ) . ThoughM8Eo(η2)(-)'S a r et h esamef o r8GWand8 t h cGWぅ e 伍c u l t yi nt h ec a s e so fz e r oandp e r f e c ts c o r e s wherea b i l i t ye s t i m a t i o n l a t t e rhasad i o e sn o tg i v e五n i t e8 . Ont h eo t h e rhand 8GW a r e byMLr e q u i r e df o r8cGWd MLs a v a i l a b l ei nt h e s ec a s e s . Asa d d r e s s e de a r l i e r, s i n c ec U : i ndependsonunknown80, c~巾 is n o ta v a i l a b l ei n p r a c t i c e . I no r d e rt oovercmnet h ed i 伍c u l t yananonymousr e v i e w e rs u g g e s t e da methodt h a tt a k e st h ee x p e c t a t i o nぱ 0 fc 山 U : i d e n s i t y( t h ea u t h o ri si n d e b t e dt ot h er e v i e w e rf o rt h es u g g e s t i o nanda s s o c i a t e dr e s u l t s ) .I np r a c t i c eぅ itemparametersa r et y p i c a l l yc a l i b r a t e dbyu s i n g8rv N(O1 )t o 五c a t i o n .80, i ti sq u i t er e a s o n a b l et ou s et h es t a n d a r dnormal o b t a i nt h emodeli d e n t i d e n s i t y( d e n o t e dbyゆ( 8 ) )f o rグ( 8 )a si susedf o rt h ep r i o ro ft h eB ME .Thati s, t h e e x p e c t a t i o ni sf o r m e r l yg i v e na s う う う 【 う 同 う 出 に =ω。 ( 5 . 5 ) n ( 8 ) } whichcanbeapproximatedbyu s i n gGaussianq u a d r a t u r e : 土 去 芝 山山 則合i n ( 8 ) } n ( 5 . 6 ) Aiう st h ei t hq u a d r a t u r ep o i n t, Ai i st h ei 七hq u a d r a t u r ew e i g h tandl V Ji st h e whereXi i む r 剖 a , t Ul町 p o i n t s( s e e8troud& 8 e c h r e s t1 9 6 6T a b l eF i v e, pp.217 numbero ft h equa d 2012b, P a r tA, 8 u b s e c t i o n3 . 1 ) . 2 5 2 ;Ogasawara, 01 5and2 0 . However, ( 5 . 6 )becomes Thea u t h o rt r i e dt h i smethodwithM = 1 ぅ・ g r e a t e rthan50, andt e n d st obeu n s t a b l e .Thisi sduet ot h el a r g e r e l a t i v e l yl a r g ee . g o ri n f i 叫t e l yl a r g ev a l u eo fc : U i n( 8 )with8b e i n garoundo .Theinstabilitycomesfrom ( 5 . 6 )becomes t h ef a c tt h a twhenaq u a d r a t u r ep o i n thappenst obec l o s et ot h i sv a l u e, l a r g e . 80ぅ somemethodst oremovet h ei n s t a b i l i t ya r er e q u i r e d .A s i m p l emethodi s : U i n ( O )s i n c e0i st h emeano fN(O, 1 )thought h i sa g a i ng i v e sal a r g ev a l u e t ou s ec U :i n( -1) ( r e c a l lT a b l e1 ) .Anothers i m p l emethodi st ot a k et h emeano rminimumo fc andc U : i n( 1 )s i n c e土 1a r evah 悶 awayfromt h emeanbyas t a n d a r dd e v i a t i o n .When wehave3 . 9 8rv6. 46fromT able1 . Whensomei n f o r m a t i o n t h eminimumi sused, aboutt h ep o s s i b l ev a l u e so fc~巾 is a v a i l a b l e amethods i n u l a rt oぅ b u td i f f e r e n tfrom う う う う ESTIMATIONOFABILITYUSINGPSEUDOCOUNTSI NITEMRESPONSETHEORY 1 4 5 thatsuggestedbytherevieweri stousetheh i e r a r c h i c a lBayesianmode l .Inthe五rst ; ; : l i ni sseenas a power ofa p r i o r( s e e( 2 . 3 ) ) . Then,usi時 thehigher-order stageぅ C n : n : p r i o rrepresentingtheinformationa v a i l a b l ef o rc i n 'c i ni sintegratedoutoverthe d i s t r i b u t i o n, whichcanavoidthei n s t a b i l i t ymentionedabove. Alternatively ,we are tempted to use estimates of80 e.g ,・ 8ML. Howeverぅ this 由 n : i sdangerousbecausetheformulaofMSE )forc i o(η2)(・ ω M 白 L L L 註 1 i n ゆ ( 九 6 C 4 4 仏 C L L 4 i 泊 n( 伊 向 0 削 0 ω )=O p(η一1 ν / 2 勺 ) . 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