Estimates and Standard Errors for Ratios of Normalizing Constants from Multiple Markov Chains via Regeneration

Transcription

1 Esimaes ad Sadard Errors for Raios of Normaizig Cosas from Muipe Markov Chais via Regeeraio Hai Doss 1 ad Aixi Ta 2 1 Deparme of Saisics, Uiversiy of Forida 2 Deparme of Saisics, Uiversiy of Iowa Absrac I he cassica biased sampig probem, we have k desiies π 1 ),..., π k ), each kow up o a ormaizig cosa, i.e. for = 1,..., k, π ) = ν )/m, where ν ) is a kow fucio ad m is a ukow cosa. For each, we have a iid sampe from π, ad he probem is o esimae he raios m /m s for a ad a s. This probem arises frequey i severa siuaios i boh frequeis ad Bayesia iferece. A esimae of he raios was deveoped ad sudied by Vardi ad his co-workers over wo decades ago, ad sice he here has bee much subseque work o his probem from may differe perspecives. I spie of his, here are o rigorous resus i he ieraure o how o esimae he sadard error of he esimae. I his paper we prese a cass of esimaes of he raios of ormaizig cosas ha are appropriae for he case where he sampes from he π s are o iid sequeces, bu are Markov chais. We aso deveop a approach based o regeeraive simuaio for obaiig sadard errors for he esimaes of raios of ormaizig cosas. These sadard error esimaes are vaid for boh he iid case ad he Markov chai case. Key words ad phrases: Geomeric ergodiciy, imporace sampig, Markov chai Moe Caro, raios of ormaizig cosas, regeeraive simuaio, sadard errors. Research suppored by NSF Gras DMS ad DMS

2 1 Iroducio The probem of esimaig raios of ormaizig cosas of uormaized desiies arises frequey i saisica iferece. Here we meio hree isaces of his probem. I missig daa or ae variabe) modes, suppose ha he daa is X obs, ad he ikeihood of he daa is difficu o wrie dow bu X obs ca be augmeed wih a par X mis i such a way ha he ikeihood for X mis, X obs ) is easy o wrie. I his case usig geeric oaio) we have p θ X mis X obs ) = p θ X mis, X obs )/p θ X obs ). The deomiaor, i.e. he ikeihood of he observed daa a parameer vaue θ, is precisey a ormaizig cosa. For he purpose of carryig ou ikeihood iferece, if θ 1 is some referece vaue, kowedge of og p θ X obs )/p θ1 X obs ) ) is equivae o kowedge of ogp θ X obs )): for hese wo fucios he maximum occurs a he same poi, ad he egaive secod derivaive a he maximum i.e. he observed Fisher iformaio) is he same. A secod exampe arises whe he ikeihood has he form p θ x) = g θ x)/z θ, where g θ is a kow fucio. This siuaio arises i expoeia famiy probems, ad excep for he usua exbook exampes, he ormaizig cosa is aayicay iracabe. If for some arbirary poi θ 1 we kow he raio z θ /z θ1, he we woud kow p θ x) up o a muipicaive cosa ad, as before, his woud be equivae o kowig p θ x) isef. A hird exampe arises i cerai hyperparameer seecio probems i Bayesia aaysis. Suppose ha we wish o choose a prior from he famiy {π h, h H}, where he π h s are desiies wih respec o a domiaig measure µ. For ay h H, he margia ikeihood of he daa X whe he prior is π h is give by m h X) = p θ X)π h θ) µdθ), i.e. i is he ormaizig cosa i he saeme he poserior is proporioa o he ikeihood imes he prior. The empirica Bayes choice of h is by defiiio argmax h m h X). Suppose ha h 1 is some arbirary poi i H. As i he previous wo exampes, for he purpose of fidig he empirica Bayes choice of h, kowig m h X)/m h1 X) is equivae o kowig m h X). Oe may aso be ieresed i he cosey reaed probem of esimaig he poserior expecaio of a fucio fθ) whe he hyperparameer is h, which is give by E h fθ) X) = fθ)p θ X)π h θ) µdθ) ) /m h X). Esimaig E h fθ) X) as h varies is reeva i Bayesia sesiiviy aaysis. The scheme for doig his used i Bua ad Doss 2011) does o ivove esimaig m h X) isef ad requires oy esimaig m h X)/m h1 X) for some fixed h 1 H.) Now, esimaio of a ormaizig cosa is geeray a difficu probem; for exampe, he so-caed harmoic mea esimaor proposed by Newo ad Rafery 1994) ypicay coverges a a rae ha is much sower ha Woper ad Schmider, 2011). O he oher had, esimaig a raio of ormaizig cosas ypicay ca be doe wih a -cosise esimaor. To iusrae his fac, cosider he secod of he probems described above, ad e µ be he measure wih respec o which he p θ s are desiies. Suppose ha X 1, X 2,... are a sampe from p θ1 iid sampe or ergodic Markov chai oupu). For he simpe ad we-kow esimaor 1/) g θx i )/g θ1 X i ) we have 1 g θ X i ) g θ1 X i ) gθ x) g θ1 x) p θ 1 x) µdx) = z θ z θ1, 1.1) ad uder cerai mome codiios o he raio g θ X i )/g θ1 X i ) ad mixig codiios o he 1

3 chai, he esimae o he ef of 1.1) aso saisfies a cera imi heorem CLT). I fac, i a he probems meioed above, i is o ecessary o esimae he ormaizig cosas hemseves, ad i is sufficie o esimae raios of ormaizig cosas. The esimaor above does o work we if θ is o cose o θ 1, or more precisey, if g θ ad g θ1 are o cose. I is beer o choose θ 1,..., θ k appropriaey spread ou i he parameer space Θ, ad o he ef side of 1.1) repace g θ1 wih s=1 w sg θs, where w s > 0, s = 1,..., k. The hope is ha g θ wi be cose o a eas oe of he g θs s, ad so precude havig arge variaces. To impeme his, suppose we kow a he raios z θs /z θ, s, {1,..., k}, or equivaey, we kow z θ1 /z θs, s {1,..., k}. I his case, if for each = 1,..., k here is avaiabe a sampe X ) 1,..., X ) 1 from g θ /z θ, he eig = ad a = /, we have a g θ X ) i ) s=1 a sg θs X ) i )z θ1 /z θs ) = a g θ x) g θ x) s=1 a µdx) 1.2) sg θs x)z θ1 /z θs ) z θ a g θ x)/z θ k s=1 a sg θs x)z θ1 /z θs ) g θx) µdx) = z θ z θ1. Whe compared wih he esimae o he ef side of 1.1), he esimae o he ef side of 1.2) is accurae over a much bigger rage of θ s. Bu o use i, i is ecessary o be abe o esimae he raios z θ1 /z θs, s {1,..., k}, ad i is his probem ha is he focus of his paper. We ow sae expiciy he versio of his probem ha we wi dea wih here, ad we chage o he oaio ha we wi use for he res of he paper. We have k desiies π 1,..., π k wih respec o he measure µ, which are kow excep for ormaizig cosas, i.e. we have π = ν /m, where he ν s are kow fucios ad he m s are ukow cosas. For each we have a Markov chai Φ = {X ) 1,..., X ) } wih ivaria desiy π, he k chais are idepede, ad he objecive is o esimae a possibe raios m i /m j, i j or, equivaey, he vecor d = m 2 /m 1,..., m k /m 1 ). Whe he sampes are iid sequeces, his is he biased sampig probem iroduced by Vardi 1985), which coais exampes ha differ i characer quie a bi from hose cosidered here. Suppose we are i he iid case, ad cosider he pooed sampe S = { X ) i, i = 1,...,, = 1,..., k }. Le x S, ad suppose ha x came from he h sampe. If we preed ha he oy hig we kow abou x is is vaue, he he probabiiy ha x came from he h sampe is π x) s=1 sπ s x) = a ν x)/m s=1 a sν s x)/m s := λ x, m), 1.3) where m = m 1,..., m k ). Geyer 1994) proposed o rea he vecor m as a ukow parameer ad o esimae i by maximizig he quasi-ikeihood fucio L m) = k λ X ) i, m) 1.4) wih respec o m. Acuay, here is a o-ideifiabiiy issue regardig L : for ay cosa c > 0, L m) ad L cm) are he same. So we ca esimae m oy up o a overa muipicaive 2

4 cosa, i.e. we ca esimae oy d. Accordigy, Geyer 1994) proposed o esimae d by maximizig L m) subjec o he cosrai m 1 = 1. A more deaied discussio of he quasiikeihood fucio 1.4) is give i Secio 2.) I fac, he resuig esimae, ˆd, was origiay proposed by Vardi 1985), ad sudied furher by Gi, Vardi ad Weer 1988), who showed ha i is cosise ad asympoicay orma, ad esabished is opimaiy properies, a uder he assumpio ha for each = 1,..., k, X ) 1,..., X ) is a iid sequece. Geyer 1994) exeded he cosisecy ad asympoic ormaiy resu o he case where he k sequeces X ) 1,..., X ) are Markov chais saisfyig cerai mixig codiios. The esimae was rederived i Meg ad Wog 1996), Kog e a. 2003), ad Ta 2004) from compeey differe perspecives, a uder he iid assumpio. As meioed earier, for he kids of probems we have i mid he disribuios π are aayicay iracabe, ad esimaes of he sor 1.1) or 1.2), ad he esimae of d are appicabe o a much arger cass of probems if we are wiig o use Markov chai sampes isead of iid sampes. Esimaio of he asympoic covariace marix of ˆd is he difficu for wo reasos. Firs, he esimae ˆd is obaied as he souio o a cosraied opimizaio probem, ad secod, whe he sequeces X ) 1,..., X ) are Markov chais isead of iid sequeces, he asympoic covariace marix has a compex form ad is difficu o esimae cosisey. The prese paper deas wih wo issues. Firs, oe of he auhors cied above give cosise esimaors of he variace, eve i he iid case. For he iid case, Kog e a. 2003) give a esimae ha ivoves he iverse of a cerai Fisher iformaio marix, bu his forma cacuaio does o esabish cosisecy of he esimae, or eve he ecessary CLT, or do he auhors make such caims.) As meioed earier, he probem of esimaig he variace is far more chaegig whe he sampes are Markov chais as opposed o iid sequeces. I his paper we give a CLT for he vecor ˆd based o regeeraive simuaio. The mai beefi of his resu is ha i gives, esseiay as a free by-produc, a simpe cosise esimae of he covariace marix i he Markov chai seig. Secod, he esimae obaied by he afore-meioed auhors is opima i he case where he sampes are iid. Whe he sampes are Markov chais, he esimaes are o oger opima. We prese a mehod for formig esimaors which are suiabe i he Markov chai seig. The regeeraio-based CLT ad esimae of he variace boh appy o he cass of esimaors ha we propose. The res of his paper is orgaized as foows. I Secio 2 we use ideas from regeeraive simuaio o deveop a CLT for ˆd, ad we show how our esimae of variace emerges as a byproduc. I Secio 3 we describe a cass of esimaors of d which are suiabe whe he sampes are Markov chais, as opposed o iid sampes, ad we aso propose a mehod for choosig a esimaor from his cass. I Secio 4 we prese a sma sudy ha iusraes he gais obaied from usig a esimae of d desiged for Markov chais, ad we iusrae our mehodoogy by showig how i ca be used o esimae cerai quaiies of ieres i he Isig mode of saisica mechaics. The Appedix provides proofs of he hree asserios made by he heorem i Secio 2, amey srog cosisecy of ˆd, he CLT for ˆd, ad srog cosisecy of he esimae of variace of ˆd. 3

5 2 A Regeeraio-Based CLT ad Variace Esimae We begi by cosiderig more carefuy he quasi-ikeihood fucio for m give by 1.4), ad for he echica deveopme i is much more coveie o work o he og scae. So defie he vecor ζ by ζ = ogm ) + oga ), for = 1,..., k, 2.1) ad rewrie 1.3) as p x, ζ) = ν x)e ζ s=1 ν sx)e ζs, for = 1,..., k. 2.2) Ceary, ζ deermies ad is deermied by m 1,..., m k ), ad he og quasi-ikeihood fucio for ζ is ζ) = og p X ) i, ζ) ). 2.3) I 2.1), m 1,..., m k ) is a arbirary vecor wih sricy posiive compoes, i.e. m eed o correspod o he ormaizig cosa for ν. We wi use ζ ) o deoe he rue vaue of ζ, i.e. he vaue i akes whe he m s are he ormaizig cosas for he ν s. The o-ideifiabiiy issue ow is ha for ay cosa c R, ζ) ad ζ + c1 k ) are he same here, 1 k is he vecor of k 1 s), so we ca esimae ζ ) oy up o a addiive cosa. Accordigy, wih ζ 0 R k defied by [ζ 0 ] = [ζ ) ] s=1 [ζ ) )] s /k, Geyer 1994) proposed o esimae ζ0 by ˆζ, he maximizer of subjec o he iear cosrai ζ 1 k = 0, ad hus obai a esimae of d. The erm p x, ζ) i 2.2) has he appearace of a ikeihood raio, ad i he deomiaor, he probabiiy measure ν s /m s is give weigh proporioa o he egh of he chai Φ s. Now Gi e a. s 1988) opimaiy resu does o appy o he Markov chai case, i which he chais Φ 1,..., Φ k mix a possiby differe raes, ad he a s s shoud i some sese refec he vague oio of effecive sampe sizes of he differe chais. The opima choice of he vecor a = a 1,..., a k ) is very difficu o deermie heoreicay, ad i Secio 3 we describe a empirica mehod for choosig a. Accordigy i 2.1) ad heceforh, a wi o ecessariy be give by a = /, bu wi be a arbirary probabiiy vecor saisfyig he codiio ha a > 0 for = 1,..., k. 2.1 Regeeraio ad a Miorizaio Codiio We are ieresed i obaiig a sadard error esimae for ˆζ. To describe our approach, we firs briefy review he avaiabe mehods for esimaig variaces based o Markov chai oupu. Because ˆζ is a compicaed esimae, we firs discuss he much simper case where we have a sige Markov chai X 1, X 2,... o he measurabe space X, B), wih ivaria disribuio π, f : X R is a fucio, ad we are ieresed i esimaig he variace of f := 1 fx i). The commoy used approaches are hose based o specra mehods, bachig, ad regeeraio see, e.g., Geyer, 1992; Mykad, Tierey ad Yu, 1995; Joes e a., 2006). Amog hese hree, he ceaes is he oe based o regeeraive simuaio. 4

6 A regeeraio is a radom ime a which a sochasic process probabiisicay resars isef. The ours made by he chai i bewee such radom imes are iid, ad his fac makes much easier he asympoic aaysis of averages, ad of saisics based o vecors of averages. I he discree sae space seig, if x X is ay poi o which he chai reurs ifiiey ofe, he he imes of reur o x form a sequece of regeeraios. For mos of he Markov chais used i MCMC agorihms, he sae space is coiuous, ad here is o poi o which he chai reurs ifiiey ofe wih probabiiy oe. Eve whe he sae space is discree, regeeraios based o reurs o a poi x, as described above, are ofe o usefu, because if x has very sma probabiiy uder he saioary disribuio, he o average i wi ake a very og ime o reur o x. Foruaey, Mykad e a. 1995) provided a geera echique for ideifyig a sequece of regeeraio imes 1 = τ 0 < τ 1 < τ 2 < ha is based o he cosrucio of a miorizaio codiio. This cosrucio wi be reviewed shory, bu we ow briefy skech how havig a regeeraio sequece {τ } =0 eabes us o cosruc a simpe esimae of he sadard error of f. Defie Y = τ 1 i=τ 1 fx i ) ad T = τ 1 i=τ 1 1 = τ τ 1, = 1, 2,..., ad oe ha he pairs Y, T ) form a iid sequece. If we ru he chai for ρ regeeraios, he he oa umber of cyces, sarig a τ 0, is give by = ρ =1 T. We may wrie f as fx i) = ρ =1 Y ρ =1 T = ρ =1 Y ) /ρ ρ =1 T. 2.4) ) /ρ Equaio 2.4) expresses f as a raio of wo averages of iid quaiies, ad his represeaio eabes us o use he dea mehod o obai boh a CLT for f ad a simpe sadard error esimae for f. A ouie of he argume is as foows. From 2.4) we see ha as ρ which impies ha ) we have E π fx)) fx i) = ρ =1 Y ) /ρ ρ =1 T ) EY 1) /ρ ET 1 ), 2.5) where he covergece saeme o he ef foows from he ergodic heorem, ad he covergece saeme o he righ foows from wo appicaios of he srog aw of arge umbers. I 2.5) he subscrip π o he expecaio idicaes ha X π.) From 2.5) we obai EY 1 ) = E π fx))et 1 ). Now he bivariae CLT gives ρ 1/2 Ȳ Eπ fx))et 1 ) T ET 1 ) ) d N 0, Σ f ), 2.6) where Σ f = Cov Y 1, T 1 ) ). The dea mehod appied o he fucio hy, ) = y/ gives he CLT ρ 1/2 Ȳ / T E π fx)) ) d N 0, σ 2 f), 5

7 where σf 2 = h) Σ f h ad h is evauaed a he vecor of meas i 2.6)). Moreover, i is sraighforward o check ha for he variace esimaor ρ ˆσ f 2 =1 = Y ft ) 2 ρ T, 2 we have ˆσ f 2 σf 2. The reguariy codiios eeded o make his argume rigorous are speed ou whe we discuss he case of he more compicaed esimaor ˆζ Secio 2.3 ad he Appedix). The argume above higes o beig abe o arrive a a sequece of regeeraio imes, ad wheher hese are usefu depeds o wheher he sequece has he propery ha he egh of he ours bewee regeeraios is o very arge. We ow describe he miorizaio codiio ha ca someimes be used o cosruc usefu regeeraio sequeces. Le Kx, A) be he Markov rasiio disribuio, ad suppose ha for each x X, Kx, ) has desiy kx, ) wih respec o a domiaig measure µ. The cosrucio described i Mykad e a. 1995) requires he exisece of a fucio s: X [0, 1), whose expecaio wih respec o π is sricy posiive, ad a probabiiy desiy q wih respec o µ, such ha k, ) saisfies kx, x ) sx)qx ) for a x, x X. This is caed a miorizaio codiio ad, as we describe beow, i ca be used o iroduce regeeraios io he Markov chai drive by k. Defie rx, x ) = kx, x ) sx)qx ). 1 sx) Noe ha for fixed x X, rx, x ) is a desiy fucio i x. We may herefore wrie kx, x ) = sx)qx ) + 1 sx))rx, x ), which gives a represeaio of kx, ) as a mixure of wo desiies, q ) ad rx, ). This provides a aeraive mehod of simuaig from k. Suppose ha he curre sae of he chai is X. We geerae δ BerouisX )). If δ = 1, we draw X +1 q; oherwise, we draw X +1 rx, ). Noe ha if δ = 1, he ex sae of he chai is draw from q, which does o deped o he curre sae. Hece he chai forges he curre sae ad we have a regeeraio. To be more specific, suppose we sar he Markov chai wih X 1 q ad he use he mehod described above o simuae he chai. Each ime δ = 1, we have X +1 q ad he process sochasicay resars isef; ha is, he process regeeraes. I pracice, simuaig from r ca be exremey difficu. Foruaey, Mykad e a. 1995), foowig Nummei 1984, p. 62), oiced a cever way of circumveig he eed o draw from r. Isead of makig a draw from he codiioa disribuio of δ give x ad he geeraig x +1 give δ, x ), which woud resu i a draw from he joi disribuio of δ, x +1 ) give x, we simpy draw from he codiioa disribuio of x +1 give x i he usua way i.e. usig k), ad he draw δ give x, x +1 ). This aeraive sampig mechaism yieds a draw from he same joi desiy, bu avoids havig o draw from r. Moreover, give x, x +1 ), δ has a Beroui disribuio wih success probabiiy give simpy by P δ = 1 x = x, x +1 = x ) = sx )qx) kx x ). 6

8 2.2 A Quasi-Likeihood Fucio Desiged for he Markov Chai Seig As meioed earier, Geyer 1994) showed ha whe we ake a j = j /, he maximizer of he og quasi-ikeihood fucio defied by 2.3) subjec o he cosrai ζ 1 k = 0) is a cosise esimae of he rue vaue ζ 0, ad aso saisfies a CLT, eve whe he k sequeces {X ) i }, = 1,..., k are Markov chais. Bu whe he k sequeces are Markov chais, he choice a j = j / is o oger opima, ad for oher choices of a, he cosraied) maximizer of 2.3) is o ecessariy eve cosise. We wi prese a ew og quasi-ikeihood fucio which does yied cosise asympoicay orma esimaes, ad before doig his, we give a brief moivaig argume. Suppose ha we are i he simpe case where we have a parameric famiy {p θ, θ Θ} iid ad we observe daa Y 1,..., Y p θ0 for some θ 0 Θ. Le y θ) = ogp θ y)), ad e Qθ) = E θ0 Y θ)). The fac ha argmax θ Qθ) = θ 0 is we kow ad easy o see via a shor argume ivovig Jese s iequaiy). The og ikeihood fucio based o Y 1,..., Y is Y i θ). By he srog aw of arge umbers, 1 Y i θ) Qθ) for a θ Θ, 2.7) ad assumig sufficie reguariy codiios, argmax θ 1 Y i θ) argmax θ Qθ) = θ 0, i.e. he maximum ikeihood esimaor is cosise. We ow reur o he prese siuaio, i which for = 1,..., k, {X ) i } is a Markov chai wih ivaria desiy π. Suppose we use ζ) give by 2.3), wih a a arbirary probabiiy vecor i.e. a is o ecessariy give by a j = j /), ad e Qζ) = E ζ0 ζ)). The key codiio argmax Qζ) = ζ 0 2.8) ζ eed o hod, ad he cosraied maximizer of ζ) may coverge, bu o o he rue vaue. Wih his i mid, suppose ha a is a arbirary probabiiy vecor wih o-zero eries ad defie w R k by w = a, = 1,..., k. 2.9) The og quasi-ikeihood fucio we wi use is ζ) = w og p X ) i, ζ) ) 2.10) isead of give by 2.3) [oe he sigh chage of oaio from o ]. As wi emerge i our proofs of cosisecy ad asympoic ormaiy of he cosraied maximizer of ζ), for his og quasi-ikeihood fucio, he sochasic process i ζ) 1 ζ) coverges amos surey o a fucio of ζ which is maximized a ζ 0, a codiio ha pays he roe of 2.7) ad 2.8). Noe ha if a = /, he w = 1 ad 2.10) reduces o 2.3). 7

9 2.3 A CLT for he Esimae Desiged for Markov Chais We assume ha for = 1,..., k, chai has Markov rasiio desiy k x, x ) wih respec o some measure µ) which saisfies he miorizaio codiio k x, x ) s x)q x ) for a x, x X 2.11) for some desiy q ad fucio s : X [0, 1) wih E π s X)) > 0, ad ha he chai has bee ru for ρ regeeraios. Le 1 = τ ) 0 < τ ) 1 < < τ ρ ) deoe he regeeraio imes of he h chai, ad e T ) = τ ) τ ) 1 be he egh of he h our of he h chai. So he egh of he h chai, = T ) T ρ ), is radom. We wi assume ha ρ 1,..., ρ k i such a way ha ρ /ρ 1 c 0, ), for = 1,..., k. We wi aow he vecor a o deped o ρ = ρ 1,..., ρ k ), i.e. a = a ρ) ahough we wi suppress his depedece i he oaio excep whe his depedece maers), ad we wi make he miima assumpio ha a ρ) α as ρ 1,..., ρ k, where α is a probabiiy vecor wih sricy posiive eries. The exra geeraiy is eeded if we wish o choose a i a daa-drive way cf. Remark 3 of Secio 3). The defiiios of ζ ad p x, ζ) give by 2.1) ad 2.2), respecivey, are si i force, ζ 0 is si he ceered versio of he rue vaue of ζ, bu ow ˆζ is he cosraied maximizer of he ew og quasi-ikeihood fucio 2.10). We wi show ha ˆζ is a cosise asympoicay orma esimae of ζ 0, ad sice ζ 0 deermies ad is deermied by d, his wi produce a correspodig esimae ˆd of d. Before proceedig, we meio he fac ha difficuies arise if he suppors of he disribuios π 1,..., π k differ he difficuies are pervasive: for he case where we have a coiuum of disribuios {π θ, θ Θ}, he simpe esimae 1.1) is o eve defied if π θ is o absouey coiuous wih respec o π θ1 ). So for he res of his paper, we wi assume ha he k disribuios π 1,..., π k are muuay absouey coiuous. We do o reay eed o make a assumpio his srog, bu he assumpio is saisfied for a he casses of probems we are cosiderig, ad makig i eimiaes some echica issues. I order o sae our CLT for he vecor ρ 1/2 1 ˆd d), we eed o defie he quaiies ha go io he expressio for he asympoic variace. We firs cosider he vecor ρ 1/2 1 ˆζ ζ 0 ), whose covariace marix is siguar sice his vecor sums o 0). The asympoic disribuio of ρ 1/2 1 ˆζ ζ 0 ) ivoves he marices B ad Ω defied beow. Le ζ α be he vecor whose compoes are [ζ α ] = ogm ) + ogα ), ad e B be he k k marix give by B rr = α j E πj pr X, ζ α )[1 p r X, ζ α )] ), r = 1,..., k, j=1 B rs = α j E πj pr X, ζ α )p s X, ζ α ) ), r, s = 1,..., k, r s. j=1 We wi be usig he aura esimae defied by 1 B rr = a p r X ) i, ˆζ) [ 1 p r X ) i, ˆζ) ]), r = 1,..., k, 1 ) B rs = a p r X ) i, ˆζ)p s X ) i, ˆζ), r, s = 1,..., k, r s ) 2.13)

10 Le y r,) i a) = p r X ) i, ζ 0 ) E π pr X, ζ 0 ) ), i = 1,...,, y r,) i α) = p r X ) i, ζ α ) E π pr X, ζ α ) ), i = 1,...,, 2.14) ad oe ha boh y r,) i a) ad y r,) i α) have mea 0. Defie Y r,) α) = τ ) 1 Y r,) a) = i=τ ) 1 τ ) 1 i=τ ) 1 y r,) i α), Ȳ r,) α) = 1 ρ y r,) i a), Ȳ r,) a) = 1 ρ ρ =1 ρ =1 Y r,) a), Y r,) α), ad T ) = 1 ρ ρ =1 T ). 2.15) Le Ω be he k k marix defied by Ω rs = To obai a esimae Ω, we e ad defie Ω by Z r,) = α 2 r,) E Y 1 α)y s,) 1 α) ) c ) ET 1 ) ) 2, r, s = 1,..., k, 2.16) τ ) 1 i=τ ) 1 p r X ) i, ˆζ) ad ˆµ ) r = p rx ) i, ˆζ), Ω rs = a c T ) ) 2 ρ ρ =1 Z r,) T ) ) ˆµ ) s,) r Z T ) ) ˆµ ) r, r, s = 1,..., k. 2.17) The fucio g : R k R k 1 ha maps ζ 0 io d is e ζ 1 ζ 2 a 2 /a 1 e ζ 1 ζ 3 a 3 /a 1 gζ) =., 2.18) e ζ 1 ζ kak /a 1 ad is gradie a ζ 0 i erms of d) is d 2 d 3... d k d D = 0 d ) d k 9

11 We have d = gζ 0 ), ad by defiiio ˆd = gˆζ). The heorem beow has hree pars, peraiig o he srog cosisecy of ˆd, asympoic ormaiy of ˆd, ad a cosise esimae of he asympoic covariace marix of ˆd. For cosisecy we eed oy miima assumpios o he Markov chais Φ 1,..., Φ k, amey he so-caed basic reguariy codiios irreducibiiy, aperiodiciy ad Harris recurrece) ha are eeded for he ergodic heorem Mey ad Tweedie, 1993, Chaper 17). CLTs ad associaed resus aways require a sroger codiio, ad he oe ha is mos commoy used is geomeric ergodiciy. The heorem refers o he foowig codiios, which perai o each = 1,..., k. A1 The Markov chai {X ) 1, X ) 2,...} saisfies he basic reguariy codiios. A2 The Markov chai {X ) 1, X ) 2,...} is geomericay ergodic. A3 The Markov rasiio desiy k saisfies he miorizaio codiio 2.11). For a square marix C, C wi deoe he Moore-Perose iverse of C. Theorem 1 Suppose ha for each = 1,..., k, he Markov chai {X ) 1, X ) 2,...} has ivaria disribuio π. 1. Uder A1, he og quasi-ikeihood fucio 2.10) has a uique maximizer subjec o he cosrai ζ 1 k = 0. Le ˆζ deoe his maximizer, ad e ˆd = gˆζ). The as ρ1, ˆd d. 2. Uder A1 ad A2, as ρ 1, ρ 1/2 1 ˆd d) d N 0, W ) where W = D B ΩB D. 2.20) 3. Assume A1 A3. Le D be he marix D i 2.19) wih ˆd i pace of d, ad e B ad Ω be defied by 2.13) ad 2.17), respecivey. The, Ŵ := D B Ω B D is a srogy cosise esimaor of W. 3 Choice of he Vecor a As meioed earier, he og quasi-ikeihood ha has bee proposed ad sudied i he ieraure ivoves he fucios p x, ζ) give by 2.2), which have he form s=1 ν x)/m, 3.1) s ν s x)/m s where i he deomiaor of 3.1), he probabiiy desiy ν s x)/m s is give weigh proporioa o he egh of he s h chai. Iuiivey, oe woud wa o repace s wih he effecive sampe size for chai s, so ha if chai s mixes sowy, he weigh ha is give o ν s x)/m s is sma. Uforuaey, here is reay o such hig as a effecive sampe size because he effec of sow mixig varies quie a bi wih he fucio whose mea is beig esimaed. Therefore, i is beer o ake a direc approach ha ivoves repacig he vecor 1 /,..., k /) by a probabiiy vecor 10

12 a, ad choose a o miimize he variace of he resuig esimaor. I shoud be emphasized ha he esimaor is a compicaed fucio of k chais.) I more deai, we do he foowig. Le S k = {a R k : a 1,..., a k 0 ad s=1 a s = 1} be he k-dimesioa simpex. For each a S k, i 3.1) repace s / by a s ad form he correspodig og quasi-ikeihood fucio see equaio 2.10)), ca i a) ζ). We e ˆζ a be he cosraied maximizer of a) ζ), ad e ˆd a be he correspodig esimae of d. Le W a be he covariace marix of ˆd a give by Par 2 of Theorem 1, ad e Ŵa be is esimae. We choose a o miimize raceŵa) his correspods o he cassica A-opima desig ). I shoud be oed ha we are abe o carry ou his opimizaio scheme precisey because Theorem 1 eabes us o esimae W a. I is aso worh oig ha oce we have cosruced he k regeeraio sequeces τ ) 0 < τ ) 1 < < τ ) i he compuaio of Ŵa for a a S. Remarks ρ, = 1,..., k, hese same sequeces may be used 1. I is aura o ask wheher i he Markov chai case our procedure gives rise o a opima esimae of d, ad we ow address his quesio. To keep he discussio as simpe as possibe, we cosider he case k = 2. Le B be he se of a bridge fucios β : X R saisfyig he codiios ha 0 < βx)π 1 x)π 2 x) µdx) < ad βx) = 0 whe eiher π 1 x) = 0 or π 2 x) = 0. I is easy o see ha whe he wo sequeces X ) 1,..., X ), = 1, 2 are each iid, for ay β B, he esimae ˆd 2 = βx1) i )ν 2 X 1) i ) 2 βx2) i )ν 1 X 2) i ) is a cosise ad asympoicay orma esimae of d 2. Meg ad Wog 1996) show ha wihi B, he fucio for which he asympoic variace is miimized is β op,iid x) = [ s 1 ν 1 x) + s 2 ν 2 x)/d 2 ] 1, where s j = j /, j = 1, 2. Because his fucio ivoves he ukow d 2, Meg ad Wog 0) 1996) propose a ieraive scheme i which we sar wih, say, ˆd 2 = 1, ad a sage m, we form 1 1 ν 2 X 1) i ) ˆd m+1) 1 s 1 ν 1 X 1) i ) + s 2 ν 2 X 1) m) i )/ ˆd 2 2 = 1 2. ν 1 X 2) i ) 2 s 1 ν 1 X 2) i ) + s 2 ν 2 X 2) m) i )/ ˆd 2 They show ha im m ˆdm) 2 exiss ad is exacy equa o he esimae cosidered by Geyer 1994), ad so esabished a equivaece bewee he ieraive bridge esimaor ad he esimae based o maximizaio of he og quasi-ikeihood fucio. Whe he sequeces X ) 1,..., X ), = 1, 2 are Markov chais, he opima bridge fucio has he form β op,mcmc x) = β x)β op,iid x), where he correcio facor, β x), is he souio o a compicaed Fredhom iegra equaio Romero, 2003) ad refecs he depedece 11

13 srucure of he wo chais. I paricuar, for he case of Markov chais, he opima bridge fucio eed o have he form βx) = [ 1 ν 1 x) + 2 ν 2 x) ] 1, 3.2) for ay 1, 2. Uforuaey, β is very hard o ideify, e aoe esimae. To cocude, sice our procedure is, effecivey, searchig wihi he cass 3.2), i wi o yied a opima esimae i geera, ad isead shoud be viewed as a mehod for yiedig esimaes which are pracicay usefu, eve if o opima. 2. A crude way o fid â op := argmi a raceŵa) is o cacuae raceŵa) as a varies over a grid i S k ad he fid he miimizig a. This is iefficie ad uecessary, as here exis efficie agorihms for miimizig rea-vaued fucios of severa variabes; see, e.g., Rober ad Casea 2004, Chaper 5). 3. The vecor â op ca be cacuaed from a sma pio experime, afer which ew chais are ru ad used o form he og quasi-ikeihood fucio âop) ζ), from which we obai ˆζ ad hece ˆd). 4. If for each, X ) 1,..., X ) is a iid sequece, he a regeeraio occurs a each sep. I his case, here is o eed o esimae a, sice he opima vaue is kow o be a j = / Meg ad Wog, 1996). The w s i 2.9) reduce o 1, ad he og quasi-ikeihood fucio 2.10) reduces o exacy he og quasi-ikeihood fucio used by Geyer 1994), so our esimae is exacy he esimae iroduced by Vardi 1985), who worked i he iid seig. 4 Iusraios Here we have wo goas. I Secio 4.1 we provide a simuaio sudy o show he gais i efficiecy ha are possibe if we use he mehod for choosig he weigh vecor a described i Secio 3. Our iusraio ivoves oy probems. The purpose of Secio 4.2 is o demosrae he appicabiiy of our mehodoogy, ad we reur o he secod of he hree casses of probems we discussed i Secio 1, where we have a famiy of probabiiy desiies of he form p θ x) = g θ x)/z θ, which are iracabe because he ormaizig cosa z θ cao be compued i cosed form. Our focus here is a bi differe, i ha we are o ieresed i esimaig he famiy z θ, θ Θ; raher, we are ow ieresed i esimaig a famiy of expecaios of he form E θ UX)), θ Θ, where U is a fucio, as we as esimaig fucios of hese expecaios. Our iusraio is i he coex of he Isig mode of saisica physics, ad we show how o esimae he iera eergy ad specific hea of he sysem as a fucio of emperaure. 4.1 Gais i Efficiecy Whe Usig he Opima Weigh Vecor a Reca ha â op = argmi a raceŵa) is cacuaed from a sma pio experime. Le ˆdâop be he correspodig esimae of d. Aso, e ˆd cov deoe he esimae of d obaied whe we use he coveioa choice a j = j /. I his secio we demosrae hrough a simuaio sudy 12

14 ha sigifica gais i efficiecy are possibe if we use ˆdâop isead of ˆd cov i siuaios where he Markov chais mix a differe raes. We cosider a very simpe siuaio where k = 2, so ha d is jus d 2. We ake π 1 ad π 2 o be wo disribuios, specificay π 1 = 5,1 ad π 2 = 5,0, where r,µ deoes he disribuio wih r degrees of freedom, ceered a µ. The represeaio π = ν /m is ake o be rivia: ν = π ad m = 1 for = 1, 2. So d 2 = m 2 /m 1 is kow o be 1, bu we proceed o esimae i as if we did kow ha fac. I our simuaios, chai 1 is a iid sequece from π 1. Chai 2 is a idepedece Meropois- Hasigs IMH) chai wih proposa desiy 5,µ. Tha is, if he curre sae of he chai is x, a proposa Y 5,µ is geeraed; he chai moves o Y wih accepace probabiiy mi { [ 5,0 Y ) 5,µ x)]/[ 5,0 x) 5,µ Y )], 1 }, ad says a x wih he remaiig probabiiy. We wi e µ rage over a fie grid i 3, 3). Noe ha whe µ = 0, he proposa is aways acceped, ad he chai is a iid sequece from 5,0, bu as µ moves away from 0 i eiher direcio, proposas are ess ikey o be acceped, ad he mixig rae of he chai is sower. I is simpe o check ha if x 5,µ x)/ 5,0 x) ) > 0, which impies ha he IMH agorihm is uiformy ergodic Megerse ad Tweedie, 1996, Theorem 2.1) ad hece geomericay ergodic. Moreover, Mykad e a. 1995, Secio 4.1) have show ha for IMH chais here is aways a scheme for producig miorizaio codiios ad regeeraio sequeces, ad here we use he scheme hey described. Our simuaio sudy is carried ou as foows. For each vaue of µ, we coduc a pio sudy o cacuae â op, usig he mehod described i Secio 3. The pio sudy is based o 1000 iid draws from π 1 ad a umber of regeeraios of he IMH Markov chai for π 2 ha gives a sampe of approximaey he same size. The we ru he mai sudy, i which we form ˆdâop where â op is obaied i he pio sudy), ad aso form ˆd cov. The mai sudy is 10 imes as arge as he pio sudy. For each µ, he above is repicaed 500 imes, ad from hese repicaes we form he sampe variace of he ˆdâop s, he sampe variace of he ˆd cov s, ad form he raio, which we ake as a measure of he efficiecy of ˆdâop vs. ˆd cov. Figure 1 gives a po of he esimae of Var ˆd cov )/ Var ˆdâop ) as µ varies over 3, 3), aog wih 95% cofidece bads, vaid poiwise he bads are cosruced via he dea mehod appied o he fucio fo, c) = o/c). From he figure we see ha, as expeced, he efficiecy is abou 1 whe µ is ear 0. Bu i grows rapidy as µ moves away from 0 i eiher direcio, reachig abou 17 whe µ is 3 or 3, ad i is reasoabe o beieve ha he efficiecy is ubouded as µ or µ. Figure 2 provides a graphica descripio of he expaaio. The figure gives a po of [â op ] 1, he firs compoe of â op, as µ varies over 3, 3). Whe µ = 0, he wo.. chais are each iid sequeces, ad â op =.5,.5), so ha ˆdâop = ˆdcov. Bu whe µ moves away from 0 i eiher direcio, chai 2 mixes more sowy, ad [â op ] 1 icreases owards 1, so ha i he erm 2.2) i our quasi-ikeihood fucio, ess weigh is give o chai 2, which is why ˆdâop is more efficie ha is ˆd cov. Of course, because he cacuaio of ˆdâop requires a pio sudy, he compariso above coud be viewed as ufair. However, for ˆdâop o perform we a ha is required, boh i heory ad i pracice, is ha â op cosisey esimae argmi a Var ˆd a ), ad for his o occur a ha is required is ha he size of he pio sudy icrease o ifiiy. Tha is, he size of he pio sudy ca icrease o ifiiy arbirariy sowy whe compared o he size of he mai sudy so, asymp- 13

15 20 Reaive efficiecy Figure 1: Esimaed reaive efficiecy of ˆdâop vs. ˆdcov, ogeher wih 95% cofidece bads. As µ moves away from 0, he mixig rae of chai 2 sows, ad he efficiecy of ˆdâop vs. ˆdcov icreases. The horizoa ie a heigh 1 serves a referece ie. µ oicay, he amou of ime o compue ˆdâop ad ˆd cov is he same. 4.2 Esimaio of he Iera Eergy ad Specific Hea as Fucios of Temperaure i he Isig Mode We cosider he Isig mode o a c c square aice wih periodic boudary codiios. Tha is, we have a graph V, E) where V deoes he se of c 2 verices of he aice, ad E deoes he se of 2c 2 edges ha coec eares eighbors o he aice. Verices i he firs ad as rows are aso cosidered eighbors, as are verices i he firs ad as coums, so he graph resides o he orus. For each verex i V, we have a radom variabe X i akig o he vaues 1 ad 1. The radom vecor X = {X i, i V } gives he sae of he sysem, ad he sae space S coais 2 c2 saes. For x S, e Hx) = i j x ix j, where he oaio i j sigifies ha i ad j are eares eighbors. For each θ Θ := [0, ), defie a probabiiy disribuio p θ o S by p θ x) = z 1 θ exp[ θhx)], x S, where z θ = x S exp[ θhx)] is he ormaizig cosa, caed he pariio fucio i he physics ieraure, ad θ = 1/κT ), where T is he emperaure ad κ is he Bozma cosa. See, e.g., Newma ad Barkema 1999, sec. 1.2) for a overview. Impora o physiciss are he iera eergy of he sysem, defied by I θ = E pθ [HX)], θ Θ, ad he specific hea, which is he derivaive of he iera eergy wih respec o emperaure, or equivaey, C θ = κθ 2 I θ θ = κθ2{ E pθ [H 2 X)] E pθ [HX)] ) 2}, θ Θ, 14

16 Opima weigh Figure 2: The pois are he medias of he firs compoe of â op, i.e. he weigh assiged o sampe 1 i he erm 2.2) i our quasi-ikeihood fucio, over he 500 repicaios a each µ. As µ moves away from 0, he weigh give o he secod sower mixig) chai decreases o 0. µ ad ieres is focused o how hese quaiies vary wih θ. Because he size of he sae space icreases very rapidy as c icreases, excep for he case c 5, he quaiies above cao be evauaed, ad MCMC mus be used. I is simpe o impeme a Meropois-Hasigs agorihm ha radomy chooses a sie, proposes o fip is spi, ad acceps his proposa wih he Meropois-Hasigs probabiiy; however his agorihm coverges very sowy. Swedse ad Wag 1987) proposed a daa augmeaio agorihm i which bod variabes are iroduced: if i ad j are eares eighbors ad X i = X j, he wih probabiiy 1 exp θ) a edge is paced bewee verices i ad j. This pariios he sae space io coeced compoes, ad eire compoes are fipped. This agorihm coverges far more rapidy ha he sige-sie updaig agorihm, ad i is he agorihm we use here. Mykad e a. 1995, sec. 5.3) deveoped a simpe miorizaio codiio for he Swedse-Wag agorihm, ad we use i here o produce he regeeraive chais ha are eeded o esimae he famiies {I θ, θ Θ} ad {C θ, θ Θ} via he mehods of his paper. We ow cosider he probem of esimaig he famiies {I θ, θ Θ} ad {C θ, θ Θ}, ad as we wi see, he issue of obaiig sadard errors for our esimaes is quie impora. We are i he framework of he secod of he hree casses of probems meioed i Secio 1, ad he wo-sep procedure give here, described i he prese coex, is as foows: Sep 1 We choose pois θ 1,..., θ k appropriaey spread ou i he regio of Θ of ieres, ad for = 1,..., k, we ru a Swedse-Wag chai wih ivaria disribuio p θ for ρ regeeraios. Usig hese k chais, we form ˆd, he esimae of he vecor d, where d = z θ /z θ1, = 2,..., k. Sep 2 For each = 1,..., k, we geerae a ew Swedse-Wag chai wih ivaria disribuio p θ for R regeeraios, ad we use hese ew chais, ogeher wih he esimae ˆd produced i Sep 1, o esimae I θ ad C θ. 15

17 We ow describe he deais ivoved i Sep 2. Deoe he h sampe i Sep 2) by {X ) i, i = 1,..., }. For each θ Θ, defie g θ x) = exp[ θhx)] for x S. Le ad e ux) = g θ x) s=1 g θ s x), vx) = Hx)ux), ad zx) = H2 x)ux), û = ˆd ux ) i ), ˆv = ˆd vx ) i ), ad ẑ = ˆd zx ) i ). These quaiies deped o θ, bu his depedece is emporariy suppressed i he oaio.) Usig E o deoe expecaio wih respec o p θ, we have Î θ := ˆv û d E vx)) d E ux)) = z θ/z θ1 ) x S Hx)p θx) z θ /z θ1 ) x S p θx) as ρ ad R for = 1,..., k, where he covergece saeme foows from ergodiciy of he Swedse-Wag chais ad he fac ha ˆd d. Simiary, we have Ĉ θ := κθ 2 ẑ ˆv ) ) 2 C θ. û û Furhermore, Theorem 2 of Ta, Doss ad Hober 2012) deas precisey wih he asympoic disribuio of esimaes of he form Îθ ad Ĉθ, i he framework of regeeraive Markov chais. This heorem, which reies o Theorem 1 of he prese paper, saes ha if i) boh Sage 1 ad Sage 2 chais saisfy A1 A3 of he prese paper, ii) for = 1,..., k R /R 1 ad ρ /ρ 1 coverge o posiive fiie cosas, ad iii) R 1 /ρ 1 coverges o a oegaive fiie cosa, he R 1/2 1 Îθ I θ ) ad R 1/2 1 Ĉθ C θ ) have asympoicay orma disribuios, ad he heorem aso provides regeeraio-based cosise esimaes of he asympoic variaces. These are he esimaes we use i his secio. We wi appy he approach described above i wo siuaios. The firs ivoves he Isig mode o a square aice sma eough so ha exac cacuaios ca be doe. This eabes us o check he performace of our esimaors ad cofidece iervas. The secod ivoves he Isig mode o a arger aice, where cacuaios ca be doe oy hrough Moe Caro mehods. We firs cosider he Isig mode o a 5 5 aice, ad we focus o he probem of esimaig C θ, he specific hea. Figure 3 was creaed usig our mehods. The ef pae gives a po of Ĉ θ, ogeher wih 95% cofidece bads vaid poiwise), ad a po of he exac vaues. The righ pae gives he sadard error esimaes for Ĉθ. To creae he figure, we used he approach described above, wih k = 5 ad θ 1,..., θ 5 ) =.3,.4,.5,.6,.7). For each = 1,..., 5, regeeraive Swedse-Wag chais of approximae) egh 10,000 were ru for θ, based o which ˆd ad Ŵ from Theorem 1 were cacuaed. We he ra idepede chais for he same five θ vaues, for as may ieraios, o form esimaes Ĉθ o a fie grid of θ vaues ha rage from.2 o 1 i icremes of.01. The po i he righ pae was obaied from he formua i Theorem 2 of = I θ 16

18 Ta e a. 2012), ad he exac vaues of C θ were obaied usig cosed-form expressios from he physics ieraure. We meio ha Newma ad Barkema 1999, sec. 3.7) aso cosidered he probem of esimaig he specific hea for he Isig mode o a 5 5 aice. They have a po very simiar o ours, bu hey produced i by ruig a separae Swedse-Wag chai for each θ vaue o a fie grid, ad each chai is used soey for he θ vaue uder which i was geeraed. I coras, our mehod requires oy k Swedse-Wag chais, where k is fairy sma, ad a chais are used o esimae C θ for every θ. Here, we have cosidered a simpe isace of he Isig mode, he socaed oe-parameer case. I is commo o aso cosider he siuaio where here is a exera mageic fied, i which case θ has dimesio 2, ad p θ x) exp θ 1 i j x ix j + θ 2 i V x i). Ruig a separae Swedse-Wag chai for each θ i a fie subgrid i dimesio 2 becomes exremey ime cosumig, whereas our approach is easiy si workabe. Specific hea C θ C θ cof. bads SEC θ ) θ Figure 3: Esimaio of he specific hea for he Isig mode o a 5 5 aice. Lef pae gives a po of he poi esimaes ad a po of he exac vaues, as θ varies. The wo pos are visuay idisiguishabe. Aso provided are 95% cofidece bads. Righ pae gives sadard errors for Ĉθ. I our secod exampe, we cosider he Isig mode o a aice, for which exac cacuaios of physica quaiies are prohibiivey expesive, ad our ieres is ow o esimaig he iera eergy. The ef pae of Figure 4 shows a po of Îθ vs. θ as θ rages from.35 o 1.5 i icremes of.01. To form he po we carried ou he wo-sep procedure discussed earier, wih k = 5 ad referece pois θ 1,..., θ 5 ) =.65,.75,.85,.95, 1.05), ad a sampe size of 100,000 for each chai i boh seps. The ef pae aso shows 95% bads, vaid poiwise, ad he righ pae shows he esimaed sadard errors. From he po, we ca see ha he sadard errors are much arger whe θ < θ 1 =.65 ha hey are whe θ θ 1. The imporace sampig esimaes are o sabe whe we ry o exrapoae beow he owes referece θ vaue, bu we ca go we above he highes referece vaue ad si ge accurae esimaes. I is our abiiy o esimae SE s hrough regeeraio ha makes i possibe for us o deermie he rage of θ s for which we have reiabe esimaes. I fac, his rage depeds i a compicaed way o he referece pois ad he sampe sizes, ad eve for he reaivey simpe case where k = 1, he rage is o simpy a ierva ceered a θ 1. θ 17

19 Iera eergy SEI θ ) θ Figure 4: Esimaio of he iera eergy for he Isig mode o a aice. Lef pae gives esimaed vaues, ogeher wih 95% cofidece bads. Righ pae gives he correspodig sadard error esimaes. Appedix: Proof of Theorem 1 Proof of Cosisecy of ˆd We firs work i he ζ domai, ad a he very ed swich o he d domai. As meioed earier, iid i he sadard exbook siuaio i which we have X 1,..., X p θ0 where θ 0 Θ, θ) is he og ikeihood ad Qθ) = E θ0 1 θ)), he cassica proof of cosisecy Wad, 1949) is based o he observaio ha Qθ) is maximized a θ = θ 0, ad ha for each fixed θ, θ) Qθ). The covergece may be o-uiform, ad care eeds o be exercised i showig ha he maximizer of θ) coverges o he maximizer of Qθ). The prese siuaio is simper i ha he og ikeihood ad is expeced vaue are wice differeiabe ad cocave, bu is more compicaed i ha we have muipe sequeces, hey are o iid, ad we have a o-ideifiabiiy issue, so ha maximizaio is carried ou subjec o a cosrai. We wi wrie ρ isead of o remid ourseves ha he ρ s are give ad he s are deermied by hese ρ s. Aso, we wi wrie ρ X, ζ) isead of ρ ζ) whe we eed o oe he depedece of ρ ζ) o X, where X = X 1) 1,..., X 1) 1,..., X k) 1,..., X k) ) k. We defie he scaed) expeced og quasi-ikeihood by θ λζ) = a E π og[p X, ζ)] ). As ρ, we have, so 1 og p X ) i 1 ρ X, ζ) λζ) for a ζ., ζ) ) E π og[p X, ζ)] ), ad so The srucure of our proof is simiar o ha of Theorem 1 of Geyer 1994), ad he ouie of our proof is as foows. Firs, defie S = {ζ : ζ 1 k = 0}, ad reca ha ˆζ is defied o be a maximizer of ρ X, ζ) saisfyig ˆζ S. 1. We wi show ha for every X, ρ X, ζ) is everywhere wice differeiabe ad cocave i ζ. 18

20 2. We wi show ha λζ) is fiie, everywhere wice differeiabe, ad cocave. We furher show ha is Hessia marix is semi-egaive defiie, ad ha is oy u eigevecor is 1 k. 3. We wi show ha λζ 0 ) = We wi oe ha he wo seps above impy ha ζ 0 is he uique maximizer of λ subjec o he codiio ζ 0 S. 5. We wi argue ha wih probabiiy oe, for every ζ, 2 ρ X, ζ) is semi-egaive defiie, ad 1 k is is oy u eigevecor. This wi show ha ˆζ is he uique maximizer of ρ X, ζ) subjec o ˆζ S. 6. We wi cocude ha he covergece of ρ X, ζ) o λζ) impies covergece of heir maximizers ha reside i S, ha is, ˆζ ζ 0. We ow provide he deais. 1. The differeiabiiy is immediae from he defiiio of ρ see 2.10)). To show cocaviy, i is sufficie o show ha for every x, og p x, ζ) ) is cocave i ζ. Now 2 og p x, ζ) ) ζ 2 = diagp) pp ), A1) where p = p 1 x, ζ),..., p k x, ζ) ). The marix iside he pareheses o he righ side of A1) is he covariace marix for he muiomia disribuio wih parameer p, so his marix is posiive semi-defiie. 2. Firs, λζ) is fiie because λζ) 0, ad λζ) = = = )] 1 a E π [og p X, ζ) a E π [og 1 + )] ν s X) ν X) eζs ζ s ) ν s X) a E π ν X) eζs ζ og1 + a) < a for a > 0) s a e ζs ζ νs x) ν x) π x) µdx) s a e ζs ζ m s πs x) m π x) π x) µdx) <. s We ow obai he firs ad secod derivaives of λ. By a sadard argume ivovig he domiaed covergece heorem, we ca ierchage he order of differeiaio ad iegraio. If v is he vecor of egh k wih a 1 i he r h posiio ad 0 s everywhere ese, he for ay x, ay ζ, ad ay {1,..., k}, [ og p x, ζ + v/m) ) og p x, ζ) )] m = 19

21 og p x, ζ ) ) / ζ r, where ζ is bewee ζ + v/m ad ζ, ad his paria derivaive is uiformy bouded bewee 1 ad 1.) So for r = 1,..., k, we have λζ) og p X, ζ) ) ) = a E π = a r a E π pr X, ζ) ). A2) ζ r ζ r Cosider he iegrad o he righ side of A2), i.e. p r X, ζ). Is gradie is give by p r / ζ r = p r p 2 r ad p r / ζ = p r p for r, ad hese derivaives are uiformy bouded i absoue vaue by 1. Hece agai by he domiaed covergece heorem, we ca ierchage he order of differeiaio ad iegraio, ad doig his gives 2 λζ) ζ 2 r = 2 λζ) ζ s ζ r = ) pr X, ζ) a E π = ζ r ) pr X, ζ) a E π = ζ s a E π [p r X, ζ) p 2 rx, ζ)] a E π [ p r X, ζ)p s X, ζ)] for s r. Defie he expecaio operaor E P = a E. From A3) we have 2 λζ) = E P J), where J = diagp) pp, ad as before p = p 1 X, ζ),..., p k X, ζ) ). As before, J is he covariace of he muiomia, so is posiive semi-defiie, ad herefore so is E P J). A3) We ow deermie he u eigevecors of 2 λζ) which is E P J)). If 2 λζ)u = 0, he u [ 2 λζ)]u = 0, so E P u Ju) = 0. Sice J is posiive semi-defiie, i has a square roo, J 1/2. Hece E P J 1/2 u 2 ) = 0, which impies Ju = 0 [P ]-a.e. The codiio Ju = 0 [P ]-a.e. is expressed as p r X, ζ) p X, ζ)u u r ) = 0 [P ]-a.e. for r = 1,..., k, A4) ad uder our assumpio ha ν 1,..., ν k are muuay absouey coiuous, A4) impies ha u r = p X, ζ)u for r = 1,..., k. So u 1 = = u k, i.e. u 1 k. 3. To show ha λζ 0 ) = 0, we wrie λζ) ζ r = a r ζ0 a = a r = a r a r ν r x)a r /m r s=1 ν sx)a s /m s π x) µdx) a π x) s=1 a sν s x)/m s ν r x)a r /m r µdx) π r x) µdx) = For ay ζ saisfyig ζ 1 k = 0, we may wrie λζ) = λζ 0 ) + ζ ζ 0 ) λζ 0 ) ζ ζ 0) 2 λζ )ζ ζ 0 ) = λζ 0 ) ζ ζ 0) 2 λζ )ζ ζ 0 ), 20

22 where ζ is bewee ζ ad ζ 0. If ζ ζ 0, i.e. ζ ζ 0 0, he sice ζ ζ 0 ) 1 k = 0, ζ ζ 0 cao be a scaar muipe of 1 k. Hece by Sep 2, ζ ζ 0 ) 2 λζ )ζ ζ 0 ) < Ceary ρ X, ˆζ) = 0. The proof ha i) 2 ρ X, ζ) is semi-egaive defiie, ii) he oy u eigevecor of 2 ρ X, ζ) is 1 k, ad iii) ˆζ is he uique maximizer of ρ X, ζ) subjec o he cosrai ζ S, is esseiay ideica o he proof of hese asserios for λζ). 6. Sice 1 ρ X, ζ) λζ) for each ζ, covergece occurs o a dese subse of S. Aso, he fucios ivoved are a cocave i he eire space of ζ s, hece are cocave i S. Therefore, we have uiform covergece of 1 ρ X, ζ) o λζ) o compac subses of S. Uder cocaviy, his is eough o impy argmax ζ S ρ X, ζ) argmax ζ S λζ), i.e. ˆζ ζ 0. Fiay, o see ha ˆd d, we wrie ˆd d = gˆζ) gζ 0 ) = gζ ) ˆζ ζ 0 ), where ζ is bewee ˆζ ad ζ 0. The fucio g acuay depeds o a ρ), so depeds o ρ, bu he gradie gζ ) is bouded for arge ρ because ˆζ ζ 0 ad a ρ) α. Therefore ˆd d. Proof of Regeeraio-Based CLT for ˆd We begi by cosiderig ρ 1/2 1 ˆζ ζ 0 ). As i he cassica proof of asympoic ormaiy of maximum ikeihood esimaors, we expad ρ a ˆζ aroud ζ 0, ad usig he appropriae scaig facor, we ge ρ1/2 1 ρ ˆζ) ρ ζ 0 ) ) = 1 2 ρ ζ )ρ 1/2 1 ˆζ ζ 0 ), A5) where ζ is bewee ˆζ ad ζ 0. Cosider he ef side of A5), which is jus ρ 1/2 1 1 ρ ζ 0 ), sice ρ ˆζ) = 0. There are severa orivia compoes o he proof, so we firs give a ouie. 1. We show ha each eeme of he vecor 1 ρ ζ 0 ) ca be represeed as a iear combiaio of mea 0 averages of fucios of he k chais pus a vaishigy sma erm. 2. Usig Sep 1 above, we obai a regeeraio-based CLT for he scaed score vecor, via a cosideraby more ivoved versio of he mehod we used i Secio 2.1: we show ha ρ 1/2 1 1 ρ ζ 0 ) d N 0, Ω), where Ω is give by 2.16). 3. We argue ha 1 2 ρ ζ ) B ad ha 1 2 ρ ζ ) ) B, where B is defied i 2.12), usig ideas i Geyer 1994). 4. We cocude ha ρ 1/2 1 ˆζ ζ 0 ) d N 0, B ΩB ). 5. We oe he reaioships d = gζ 0 ) ad ˆd = gˆζ), where g was defied by 2.18), ad appy he dea mehod o obai he desired resu. We ow provide he deais. 21

23 1. We sar by cosiderig 1 ρ ζ 0 ). For r = 1,..., k, we have ρ ζ 0 ) ζ r = w r r 1 pr X r) i, ζ 0 ) ) r w p r X ) i, ζ 0 ) r = w r 1 p r X r) i, ζ 0 ) [ 1 E πr pr X, ζ 0 ) )]) r w [ pr X ) i, ζ 0 ) E π pr X, ζ 0 ) )] + e, A6) where e = w r r [ 1 Eπr pr X, ζ 0 ) )] r w We caim ha e = 0. To see his, oe ha from A7) we have E π pr X, ζ 0 ) ). A7) e = w r r w E π pr X, ζ 0 ) ) = w r r w a r a E πr p X, ζ 0 ) ). A8) The as equaiy i A8) hods because E π pr X, ζ 0 ) ) = = ν r x)e [ζ 0] r s=1 ν π x) µdx) = sx)e [ζ 0] s ν r x)a r /m r s=1 ν sx)a s /m s π x) µdx) ν x)a r /m s=1 ν sx)a s /m s π r x) µdx) = a r a E πr p X, ζ 0 ) ). Therefore, usig he fac ha w a r /a = w r r, we ge e = w r r w r r E π r p X, ζ 0 ) ) = w r r w r r E πr p X, ζ 0 ) ) = 0. We summarize: Because e = 0, A6) ca be used o view 1 ρ ζ 0 )/ ζ r as a iear combiaio of mea 0 averages of fucios of he k chais. To express hese averages i erms of iid quaiies, we firs reca he defiiios of y r,) i a), Y r,) a), Ȳ r,) a), ad T ), give 22

24 i 2.14) ad 2.15), ad muipyig by he scaig facor ρ 1/2 1 1, we rewrie A6) as ρ 1/2 1 ρ ζ 0 ) = ρ1/2 1 ζ r = = = = w ρ 1/2 ρ 1 w 1 =1 ρ 1/2 1 ρ w =1 Y r,) ρ =1 T ) [ ρ1 [ ρ1 ρ ) 1/2 w ρ ) 1/2 a [ pr X ) i, ζ 0 ) E π pr X, ζ 0 ) )] Y r,) a) ] [ ρ 1/2 ] [ ρ 1/2 a) ] Ȳ r,) a) T ) ] Ȳ r,) a). A9) T ) 2. We ow appy a more compex ad more rigorous versio of he argume we used i Secio 2.1. We oe he foowig: i) he k chais are geomericay ergodic by Assumpio A2; r,) ii) sice p r x, ζ) 0, 1) for a x ad a ζ, E π y 1 a) 2+ɛ) < for some ɛ > 0 i fac for ay ɛ > 0); ad iii) by 2.14) he mea of Y r,) a) is 0. The usua CLT for iid sequeces does o appy o he sequece Y r,) 1 a),..., Y ρ r,) a) because a = a ρ) is aowed o chage wih ρ, so he disribuio of Y r,) a) chages wih ρ. Sice r ad are ow fixed ad pay o impora roe, whie he depedece of a o ρ ow eeds o be oed we wi wrie y i a ρ) ) isead of y r,) i a), Y a ρ) ) isead of Y r,) a), ec. We reay have a riaguar array of radom variabes, ad we wi appy he Lideberg-Feer versio of he CLT. We firs eed o show ha E [Y a ρ) )] 2) <. This codiio is orivia because Y a ρ) ) is he sum of a radom umber of erms.) Noe ha sice p r x, ζ) 0, 1), y i a ρ) ) 1, ad herefore, Y a ρ) ) T ). A10) Theorem 2 of Hober, Joes, Prese ad Roseha 2002) saes ha E [ T ) ) 2] < uder geomeric ergodiciy. So E [Y a ρ) )] 2) <, ad we may form he riaguar array whose ρ h row cosiss of he variabes U 1 a ρ) ),..., U ρ a ρ) ), where U a ρ) ) = Y a ρ) ) ρ s=1 Var[Y sa ρ) )] ). 1/2 Ceary, E[U a ρ) )] = 0 ad ρ =1 Var[U a ρ) )] = 1. The Lideberg Codiio is ha for every η > 0, ρ =1 E [U a ρ) )] 2 I U a ρ) ) > η) ) 0 as ρ, 23