ASYMPTOTIC NORMALITY OF THE MAIMUM LIKELIHOOD ESTIMATOR IN STATE SPACE MODELS Jes Ledet Jese 1 Niels Vver Peterse 2 Uiversity of Aarhus 1;2 ad MaPhySto 1 Abstract State space models is a very geeral class of time series models capable of modelig depedet observatios i a atural ad iterpretable way. Iferece i such models have bee studied by Bickel et al., who cosider hidde Markov models, which are a special kid of state space models, ad prove that the maximum likelihood estimator is asymptotically ormal uder mild regularity coditios. I this paper we geeralize the results of Bickel et al. to state space models, where the latet process is a cotiuous state Markov chai satisfyig regularity coditios, which are fullled if the latet process takes values i a compact space. AMS 1991 subject classicatio. Primary 62F12; secodary 62M09. Keywords ad phrases. State space models, asymptotic ormality, maximum likelihood estimatio. 1. Itroductio. A state space model is a discrete time model for depedet observatios fy k g, where the depedece is modelled via a uobserved Markov process f k g such that, coditioally o f k g the Y k 's are idepedet, ad the distributio of Y k depeds o k oly. The uobserved process f k g is ofte referred to as the latet process. The state space framework ecompasses the classical ARMA models, but, more iterestigly, o-liear ad o-gaussia models ca be formulated i this framework as well. We will cosider iferece i state space models by the likelihood method. The likelihood fuctio ca ot always be calculated explicitly i these models, however, for liear state space models with Gaussia errors the likelihood fuctio ca be calculated Cetre for Mathematical Physics ad Stochastics, fuded by a grat from the Daish Natioal Research Foudatio. 1
2 J. L. Jese ad N. V. Peterse by the Kalma lter. There is a extesive literature o Kalma lterig, see for istace West & Harriso (1989) who gives a comprehesive treatmet of liear state space models with may examples. For o-liear state space models ad for state space models with o-gaussia errors the likelihood fuctio ca rarely be calculated explicitly. Istead dieret approximatios to the likelihood fuctio have bee proposed. Kitagawa & Gersch (1996) discusses a approximatio to the likelihood fuctio based o umerical itegratio techiques, a approach which is also studied i Fruhwirth-Schatter (1994). However, with these techiques the likelihood fuctio ca oly be approximated to a certai degree of accuracy. Alteratively the likelihood fuctio ca be approximated to ay degree of accuracy by simulatio techiques. This approach is ivestigated by Durbi & Koopma (1997), Shephard & Pitt (1997) ad refereces therei. Iferece i state space models has maily bee studied i the case of hidde Markov models where the latet process takes values i a ite set. Leroux (1992) proved cosistecy of the maximum likelihood estimator ad Bickel, Ritov & Ryde (1998) proved asymptotic ormality. The purpose of this paper is to exted the results of Bickel et al. to cover more geeral state space models where the latet process is a cotiuous Markov process. We show that the distributioal iequality i Lemma 4 i Bickel et al. is valid i our setup also, uder regularity coditios which ca be fullled if the state space of the latet process is a compact set. The iequality states a mixig property of the latet process, give the observed process, ad is the mai key to provig asymptotic ormality. Havig established this mixig result we follow Bickel et al. i their proof of the cetral limit theorem for the score fuctio ad i the proof of the uiform law of large umbers for the observed iformatio. I Sectio 2 we state the model ad the assumptios we will work uder. I Sectio 3 we state our mai results, the cetral limit theorem for the score fuctio, the uiform law of large umbers for the observed iformatio, ad, ally, asymptotic ormality of the maximum likelihood estimate. I Sectio 4 we prove the cetral limit theorem ad i Sectio 5 we prove the law of large umbers. 2. Notatio ad assumptios. Let f k g deote a statioary homogeous Markov chai o the measurable space ( ; A; ). Here may be cotiuous or discrete. A typical settig fulllig our assumptios below, is where is a compact set. Let (x; z) deote the trasitio desities with respect to which are parametrized by a parameter 2 R d. Let fy k g be a sequece of stochastic variables o the measurable space (Y; B; ) such that give f k g the Y k 's are idepedet, ad the distributio of Y i depeds through f k g o i oly ad has desity g (y i jx i ) wrt.. The model ca thus be formulated as Y k j k g (y k j x k ); k j k?1 (x k?1 ; x k ): We will let deote the desity wrt. of the statioary distributio of.
Asymptotics i state space models 3 We observe values Y 1 ; Y 2 ; : : : ; Y of the process fy k g while f k g remais uobserved, ad we wish to estimate by the maximum likelihood method. We will let l () deote the log likelihood fuctio based o Y 1 ; : : : ; Y. I Sectio 4 a expressio for this fuctio is derived. For the momet we oly give the expressio for the simultaeous desity of ( 1 ; : : : ; ; Y 1 ; : : : ; Y ) wrt., p (x 1 ; : : : ; x ; y 1 ; : : : ; y ) = (x 1 )g (y 1 j x 1 ) Y k=2 f (x k?1 ; x k )g (y k jx k )g: (1) Above, as everywhere else i this paper, we use the sloppy, but hopefully clear otatio p (z) for the desity of a stochastic vector with respect to a measure give by the cotext. We will let Dg deote the gradiet of g wrt. ad D 2 g will deote the Hessia, ad we will let (x) = D log (x); (x; z) = D log (x; z) ad (yjx) = D log g (yjx). The true parameter will be deoted 0 ad a otatio like 0 is short for 0. Throughout the paper 1 will deote the vector ( 1 ; : : : ; ) ad c will deote a uspecied ite costat. I the assumptios below we will let jj jj deote the max-orm of a d d maxtrix, jjajj = max i;j ja ij j. We will assume that there exists a > 0 such that with B 0 = f 2 j j? 0 j < g the followig coditios hold. (A1) There exists a > 0 ad a M < 1 such that (x; z) M for all x; z 2 ad all 2 B 0 : (A2) For all x; z 2 the maps 7! (x; z) ad 7! (x) are twice cotiuously dieretiable o B 0. Likewise, for all x 2 ad y 2 Y the map 7! g (yjx) is twice cotiuously dieretiable o B 0. (A3) Dee (y) = sup 2B0 sup x;z2 g (yjz)=g (yjx); the if x2 Y g 0 (yjx)=(y) (dy) > 0: (A4) (i) sup 2B0 sup x;z2 j (x; z)j < 1 ad sup 2B0 sup x2 j (x)j < 1. (A5) (ii) sup 2B0 sup x;z2 jjd (x; z)jj < 1 ad sup 2B0 sup x2 jjd (x)jj < 1: (iii) Dee (Y 1 ) = sup 2B0 sup x2 j (Y 1 jx)j the (Y 1 ) 2 L 2 (P 0 ) ad sup 2B0 sup x2 jjd (Y 1 jx)jj 2 L 1 (P 0 ): (i) For -almost all y 2 Y there exists a fuctio h y :! R + i L 1 () such that jg (yjx)j h y (x) for all 2 B 0. (ii) For -almost all x 2 there exist fuctios h 1 x : Y! R + ad h 2 x : Y! R + i L 1 () such that jdg (yjx)j h 1 x(y) ad jjd 2 g (yjx)jj h 2 x(y) for all 2 B 0 : (A6) 0 2 it():
4 J. L. Jese ad N. V. Peterse Remark 2.1 Note that if sup x;z2 j (x; z)j < 1 for a 2 B 0 ad sup x2 j (x)j < 1 for a 2 B 0 the A4(ii) implies A4(i). Likewise i A5(ii) the local domiated -itegrability assumptio of y 7! jjd 2 g (yjx)jj for -almost all x implies a similar property of y 7! jdg (yjx)j; provided that jdg (yjx)j 2 L 1 (P 0 ) for a 2 B 0. Remark 2.2 By assumptios A5(i), A1 ad A4 the fuctio x 1 7! D i p (x 1; y 1 ) is locally domiated -itegrable aroud 0 for -almost all y 1, ay 2 N ad i = 1; 2. This is see by otig that by (1) Dp (x 1; y 1 ) cosists of a sum of terms like, for istace, (x 1 )g (y 1 jx 1 )?1 Y k=2;k6=j f (x k?1 ; x k )g (y k jx k )g (x j?1 ; x j )Dg (y j jx j ): By A1 ad A5(i) this term is absolutely domiated by M Y k=1 h yk (x k )jd log g (y j jx j )j M sup sup 2B 0 x2 j (y j jx)j Y k=1 h yk (x k ); which for almost all xed y 1 is a -itegrable fuctio, by assumptio A4(iii). The domiatio of the secod derivative is similar. The local itegrability assumptio is eeded to \iterchage itegratio ad dieretiatio" i some expressios below. Remark 2.3 The process Y is ergodic uder A1. To see this, we observe that Y k ca be described as Y k = g( k ; U k ; 0 ), where U 1 ; U 2 ; : : : are uiformly distributed U(0; 1), ad idepedet of. Now f( k ; U k )g is a statioary Markov chai, with trasitio desity p(x 1 ; u 1 j x 0 ; u 0 ) = 0 (x 0 ; x 1 ) ad hece ergodic by A1. Thus Y is also ergodic. Remark 2.4 The assumptios A1, A3 ad A4 are restrictive ad are ot fullled i a geeral state space model. A typical example where A1 to A5 are fullled is the followig. Suppose is a compact set i R q ad is the Lebesque measure. If the trasitio desity (x; z) ad the statioary desity (x) are positive ad satises A2 ad if (x; z), (x) ad their rst ad secod derivatives are cotiuous fuctios of (; x; z) ad (; x), respectively, the A1, A4(ii) ad A4(i) are satised. Suppose, furthermore, that g (yjx) is a expoetial family desity, g (yjx) = e (x;)t(y)?((x;)) : Here deotes the cumulat trasform of t(y ) deed o the full parameter space R k, ad (x; ) 2 0 where 0 is a subset of it(). Suppose that (x; ) is twice dieretiable wrt. ad that ad its derivatives are cotiuous fuctios of (x; ), the (x; ) : B0! 0 takes values i a compact set. By cotiuity of we have g (yjx)=g (yjz) = exp h f(x; )? (z; )g t(y)? f((x; ))? ((z; ))g i c 1 exp(c 2 jt(y)j):
Asymptotics i state space models 5 The if g 0(yjx)=(y) (dy) c?1 1 if x2 Y x2 Y e(x; 0)t(y)?((x; 0)) e?c 2jt(y)j (dy) c 3 if e?j(x; 0)j jt(y)j e?c2jt(y)j (dy) x2 Y c 3 e?c4jt(y)j (dy) > 0; Y hece A3 is fullled. As for A4(iii) we have D log g (yjx) = Df(x; ) t(y)? ((x; ))g = @(x; ) ft(y)? ((x; ))g; @ T where () = @() is the mea of t(y ) uder P @ ad @(x;) deotes the d k matrix @ T of partial derivatives of wrt.. The because of compactess of B 0 we get ad hece E 0 (sup sup 2B 0 x2 sup sup 2B 0 x2 jd log g (yjx)j c 1 + c 2 jt(y)j; jd log g (yjx)j 2 ) 2c 2 1 + 2c 2 2E 0 (jt(y )j 2 ) < 1: The secod derivative D 2 log g (yjx) ca be domiated i the same way, ad hece A4 follows. Assumptio A5(i) follows agai by compactess of the parameter space, ad ally A5(ii) follows by the cotiuity of. 3. Mai results. Our mai results are stated i this Sectio. These are a cetral limit theorem for the score fuctio ad a uiform law of large umbers for the observed iformatio. As a cosequece of these we d that with a probability that teds to oe as teds to iity, there exists a (local) maximum poit of the likelihood fuctio, which is cosistet i probability ad asymptotically ormal. If especially the maximum likelihood estimator exists ad is cosistet, it is asymptotically ormal. Let l () deote the log likelihood fuctio based o observatios Y 1 ; : : : ; Y. Below, I 0 will deote a Fisher iformatio matrix give by I 0 = E 0 ( T ); where = lim!1 D log p 0 (Y 1 j Y 0?): This will be formally deed i Sectio 4, but as the followig theorems show it ca be thought of as the asymptotic covariace matrix of the score fuctio or the limit of the ormed observed iformatio as the umber of observatios teds to iity. Theorem 3.1 As teds to iity?1=2 Dl ( 0 )! N(0; I 0 ); P 0 -weakly. This Theorem is proved i Sectio 4.
6 J. L. Jese ad N. V. Peterse Theorem 3.2 Let f g deote ay stochastic sequece i such that! 0 i P 0 -probability as! 1. The?1 D 2 l ( )!?I 0 i P 0 -probability as! 1. This Theorem is proved i Sectio 5. Havig established these two results the followig result is proved i Jese (1986) (see also Sweetig (1980).) Theorem 3.3 Assume that I 0 is positive deite. With a P 0 -probability that teds to 1 as teds to iity, there exists a sequece of local maximum poits of the likelihood fuctio f^ g such that ^! 0 i P 0 probability, ad p (^? 0 )! N(0; I?1 0 ); P 0? weakly: If especially the maximum likelihood estimator exists ad is cosistet i P 0 -probability, the this estimator has the same limit distributio. The proof of the secod part of the theorem follows by a Taylor expasio of the likelihood fuctio aroud 0, as the proof of Theorem 1 i Bickel et al. The proof of the rst part relies o the assumptio that I 0 is positive deite, thus i the limit the likelihood fuctio has egative curvature ad hece a local maximum at 0. 4. A cetral limit theorem for the score fuctio. I this Sectio we prove the cetral limit theorem for the score fuctio stated i Theorem 3.1. Bickel et al. proved the same result i case where the state space of the latet process is ite. Here we start with some Lemmas which will replace Lemma 4 ad 5 i Bickel et al. For otatioal reasos we will assume that d is equal to oe i the rest of this paper. If derivatives are replaced by gradiets ad secod derivatives by Hessias all results are valid for geeral d. Lemma 4.1 Let J be a iteger set ad let 2 B 0 : Coditioally o Y J = fy j jj 2 Jg, costitute a ihomogeeous Markov chai with p ( k j k?1 ; Y J )! k, where ( 2 =(M(Y! k = k )) if k 2 J 2 =M if k 62 J: The iequality is also true for the reversed chai f?k g k2. Proof The Markov property is proved by cosiderig < k < m, assumig for simplicity that j m for all j 2 J. The Y m?1 p ( k?1 ; m j k+1 k; Y J ) = ( ) i= ( i ; i+1 ) Y j2j = h 1 ( k ; Y J)h 2 ( m k ; Y J); g (Y j j j )=p ( k ; Y J ) where h 1 ad h 2 are fuctios of ( k ; Y J) ad ( m k ; Y J), respectively. It follows that k?1 ad m k+1 are coditioally idepedet give ( k; Y J ).
Asymptotics i state space models 7 Suppose k 2 J. Coditioally o k?1 ad k+1, k ad Y Jfkg are idepedet by deitio of the state space model. Hece the coditioal desity of k give ( k?1 ; k+1 ; Y J ) is p ( k j k?1 ; k+1 ; Y J ) = ( k?1 ; k ) ( k ; k+1 )g (Y k j k ) R ( k?1 ; x) (x; k+1 )g (Y k jx) (dx) 2 M ( k?1 ; x) g (Y k jx) g (Y k j k ) (dx)!?1 2 =(M(Y k )): (2) Itegratig the coditioal desity wrt. p ( k+1 j k?1 ; Y J ) gives the stated result. Whe k 62 J the term g (Y k j k ) vaishes. The proof of the statemet for the reversed chai follows by itegratig (2) wrt. p ( k?1 j k+1 ; Y J ) istead. We state the followig Lemma for future referece, leavig the proof to the reader. Lemma 4.2 Let ( ; A; ) be a measure space ad let h:! R be a measurable fuctio o. Let 1 ad 2 be two measures domiated by with 1 ( ) = 2 ( ). The j h d 1? where S + = f d 1 d? d 2 d > 0g: h d 2 j fsup h(x)? if h(x)gf 1(S + )? 2 (S + )g; x2 x2 I the ext Lemma we will let f k g k2deote ay ihomogeous Markov chai, that is, f k g is ot ecessarily the latet process i the model. Lemma 4.3 Let f k g k2be a Markov chai with state space ( ; A; ). Assume dp k j k?1 (xjz) = p k (z; x) k ; d for all x; z 2 ad k 2, where P k j k?1 give k?1. The for ay A 2 A, deotes the coditioal distributio of k sup P ( 2 Aj 0 = )? if P ( 2 Aj 0 = ) 2 2 Y k=1 (1? k ( )): Proof Let S + = k fx 2 j p k(; x)? p k (; x) > 0g for xed ad i, ad let S? = k (S+ k )c : Dee M (k) = sup A 2 P ( 2 Aj k = ) ad m (k) = if A 2 P ( 2
8 J. L. Jese ad N. V. Peterse Aj k = ): The M (k?1) A = sup ; = sup ;? m (k?1) A (P ( 2 Aj k?1 = )? P ( 2 Aj k?1 = )) P ( 2 Aj k = z)fp k (; z)? p k (; z)g (dz) supfp ( k 2 S + k j k?1 = )? P ( k 2 S + k j k?1 = )g(m (k)? A m(k) ) A ; = supf1? P ( k 2 S? k j k?1 = )? P ( k 2 S + k j k?1 = )g(m (k) A? m (k) A ) ; f1? k ( )g(m (k) A? m (k) A ); where the rst iequality follows from Lemma 4.2. The result ow follows by iductio with k = ;? 1; : : : ; 1. (Proof based o Doob (1953, p. 198)) We are ow ready to prove a result correspodig to Lemma 4 i Bickel et al. Let!(y) = ( ) 2 =(M(y)). Lemma 4.4 Let k < l ad let J such that fk; k + 1; :::; l? 1g J. Let Y J = fy j j j 2 Jg the for ay 2 B 0, sup sup A2A ;2 jp ( k 2 A j Y J ; l = )? P ( k 2 A j Y J ; l = )j Likewise, if l < k ad fl + 1; l + 2; : : : ; kg J the sup sup A2A ;2 jp ( k 2 A j Y J ; l = )? P ( k 2 A j Y J ; l = )j l?1 Y i=k ky i=l+1 (1?!(Y i )): (1?!(Y i )): Proof we get Cosider the case k < l. Applyig Lemma 4.1 o the reversed chai f?k g k2 p ( i j i+1 ; Y J ) 2 =(M(Y i )) =!(Y i )=( ) for i = k; : : : ; l? 1: Usig Lemma 4.3 with i =!(Y i )=( ) we get the stated result. The proof is similar whe l < k, applyig Lemma 4.1 o the origial chai f k g k2.
Asymptotics i state space models 9 Lemma 4.5 Let?m? k 0. For ay i B 0 ad ay A; B 2 A we have jp ( k 2 A j Y 1? )? P ( k 2 A j Y 0? )j i=k (1?!(Y i )); jp ( k 2 A; k+1 2 B j Y 1? )? P ( k 2 A; k+1 2 B j Y 0? )j jp ( k 2 A j Y 1? )? P ( k 2 A j Y 1?m )j k Y i=? 0 Y i=k+1 (1?!(Y i )); jp ( k 2 A; k+1 2 B j Y 1? )? P ( k 2 A; k+1 2 B j Y 1?m )j k Y i=? (1?!(Y i )); (1?!(Y i )): The rst ad secod expressio hold P -almost surely if is replaced by 1. The third ad fourth hold P -almost surely if m is replaced by 1 ad for both we ca replace Y 1? ad Y 1?m by Y 0? ad Y 0?m, respectively. Proof The rst expressio ca be evaluated as jp ( k 2 A j Y 1? )? P ( k 2 A j Y 0? )j = j P ( k 2 A j Y 0?; x 1 )fp (x 1 j Y 1?)? p (x 1 j Y 0?)g (dx 1 )j sup P ( k 2 A j Y 0? ; 1 = )? if P ( k 2 A j Y 0? ; 1 = ) 2 2 i=k (1?!(Y i )); where the iequalities follows from Lemma 4.2 ad 4.4, respectively. As for the secod expressio, jp ( k 2 A; k+1 2 B j Y 1?)? P ( k 2 A; k+1 2 B j Y 0?)j = j B P ( k 2 Ajx k+1 ; Y k?)fp (x k+1 j Y 1?)? P (x k+1 j Y 0?)g (dx k+1 )j jp ( k+1 2 S + j Y 1?)? P ( k+1 2 S + j Y 0?)j i=k+1 (1?!(Y i )): Here S + is a set chose as i Lemma 4.2 ad the secod iequality follows from above. By a martigale covergece theorem by Levy (Homa-Jrgese 1994, p. 505) we get, for istace, that P ( k 2 A j Y?) 1! P ( k 2 A j Y?1) 1 P -almost surely as! 1. This result shows that we ca replace by 1 i the iequalities above. The third expressio is proved as the rst by coditioig o??1 = x??1 i the itegral, ad the fourth expressio follows from the third by a argumet similar to the oe used to deduce the secod from the rst. The argumets are idetical whe replacig Y? 1 ad Y?m 1 with Y? 0 ad Y?m 0, ad the extesio to the case m = 1 follows from the martigale covergece argumet above.
10 J. L. Jese ad N. V. Peterse Lemma 4.5 correspods to Lemma 5 i Bickel et al. Havig established this result the rest of the proof of the CLT for the score fuctio follows the lie of these authors closely. However, we will repeat some of the argumets here sice there are some diereces due to our latet process beig cotiuous. We will for otatioal reasos deote our observatios Y? ; : : : ; Y 1. The score fuctio Dl() is the give by Dl() = 1 k=? D log p (Y k j Y k?1? ); where p (Y k j Y k?1? ) deotes the coditioal desity of Y k give Y k?1? give by D log p (Y k j Y k?1? ) = D log p (Y k?)? D log p (Y k?1? ): Usig assumptio A5(i) to iterchage itegratio ad dieretiatio below we d that for ay j = k? 1; k, Hece D log p (Y k j Y k?1? ) is give by D log p (Y k j Y k?1? ) D log p (Y j?) = E (D log p (Y j?; k?) j Y j?): = E (D log p (Y k?; k?) j Y k?)? E (D log p (Y k?1? ; k?) j Y k?1? ): (3) Usig the expressio for p (Y k? ; k?) i (1) we d D log p (Y k j Y k?1? ) = k?1 i=? Now, let 1 = o E ( ( i ; i+1) + (Y i j i )jy?) k? E ( ( i ; i+1) + (Y i j i )jy k?1? ) 0 i=?1 + E ( (? ) j Y k )?? E ( (? ) j Y k?1 ) +? E ( (Y k j k ) j Y k?): (4) E0 ( 0 ( i ; i+1) + 0 (Y i j i )jy 1?1)? E 0 ( 0 ( i ; i+1 ) + 0 (Y i j i )jy 0?1) o + E 0 ( 0 (Y 1 j 1 ) j Y 1?1): (5) The iite sum is absolutely coverget i L 2 (P 0 ), as will be show i Lemma 4.6, so 1 is a well deed variable i L 2 (P 0 ). Let I 0 = E 0 ( 2 1): Lettig jj jj 2 deote the L 2 (P 0 )-orm we have:
Asymptotics i state space models 11 Lemma 4.6 There exists a 2 [0; 1) ad a costat c such that for all. Proof Let jjd log p 0 (Y 1 j Y 0?)? 1 jj 2 c ; k = 0 ( k ; k+1 ) + 0 (Y k j k ): By splittig the sums i (4) ad (5) we ca domiate jjd log p 0 (Y 1 j Y 0? )? 1jj 2 by the sum of the followig terms: jje 0 ( 0 (? ) j Y?) 1? E 0 ( 0 (? ) j Y?)jj 0 2 ; (6) jje 0 ( 0 (Y 1 j 1 ) j Y 1 )? E? 0( 0 (Y 1 j 1 ) j Y?1 1 )jj 2; (7) 0 k=?[=2] jje 0 ( k j Y j?)? E 0 ( k j Y j?1)jj 2 ; j = 0; 1; (8)?[=2]?1 k=??[=2] k=?1 jje 0 ( k j Y 1?)? E 0 ( k j Y 0?)jj 2 ; (9) jje 0 ( k j Y 1?1 )? E 0( k j Y 0?1 )jj 2; (10) where [] deotes the iteger part. We will show that each of the terms (6){(10) ca be domiated by c, where 0 < 1, which proves the Lemma. Furthermore, the domiatio of (10) shows that the sum i (5) is absolutely coverget as stated earlier. We will show the domiatio of (9) ad leave the remaiig terms to the reader. We will rst cosider the part of k give by 0 (Y k j k ) i (9). By applyig Lemma 4.2 ad 4.5 we have the followig iequality: E 0 ( 0 (Y k j k ) j Y?) 1? E 0 ( 0 (Y k j k ) j Y?) 0 = 2 sup x2 0 (Y k jx k )fp 0 (x k j Y 1? )? p 0(x k j Y 0? )g (dx k) j 0 (Y k jx)j i=k+1 Hece the L 2 -orm ca be domiated as (1?!(Y i )): jje 0 ( 0 (Y k j k ) j Y 1?)? E 0 ( 0 (Y k j k ) j Y 0?)jj 2 2 4E 0 0 @ E0 0 @ sup 0 (Y k jx) 2 x2 0 = 4E 0 @ E0 (sup 0 (Y k jx) 2 j k ) x2 i=k+1 11 (1?!(Y i )) 2 0 AA k i=k+1 E 0 ((1?!(Y i )) 2 j i ) c?k ; (11) 1 A
12 J. L. Jese ad N. V. Peterse where the equality follows by deitio of the state space model ad where is give by = sup x2 Y sup x2 = 1? Y 1? 1? ( )2 M! 2 ( )2 g 0 (yjx) (dy) M(y) ( )2 M(y) if x2 Y! g 0 (yjx) (dy) g 0 (yjx)=(y) (dy) < 1; by assumptio A3. The costat i (11) is ite by assumptio A4. For a sum of L 2 -orms we get,?[=2]?1 k=? jje 0 ( 0 (Y k j k ) j Y 1?)? E 0 ( 0 (Y k j k ) j Y 0?)jj 2?[=2]?1 c k=??k=2 c [=2]=2 : The part of (9) ivolvig 0 ( k ; k+1 ) ca be domiated i a similar way, usig A4 ad the secod iequality i Lemma 4.4. Hece we have proved the claimed domiatio of (9). Lemma 4.6 is the al brick eeded to prove Theorem 3.1; it tells us that i the limit the score fuctio fuctio is equivalet to a sum of terms like 1. These costitute a statioary martigale icremet sequece, ad hece by a martigal cetral limit theorem we obtai the stated limit distributio of the score fuctio. The proof is idetical to the proof of Lemma 1 i Bickel et al. (p. 1626) 5. A law of large umbers for the observed iformatio. I this Sectio we will show Theorem 3.2. As i the previous Sectio we will start with some Lemmas providig iequalities for coditioal probabilities. Lemmas 5.1 ad 5.3 are multivariate versios of Lemma 4.5. Lemma 5.1 Let?m? k l 0, ad let 2 B 0. The for all C 2 f( t ; Y t ) : t lg we have jp (C j Y 1?)? P (C j Y 0?)j i=l (1?!(Y i )): Likewise for all C 2 f( t ; Y t ) : t kg ad for j = 0; 1 we have, jp (C j Y j?)? P (C j Y j?m)j ky i=? (1?!(Y i )):
Asymptotics i state space models 13 Proof Let C 2 f( t ; Y t ); t lg. The jp (C j Y 1?)? P (C j Y 0?)j = j P (C j x l ; Y l?)fp (x l j Y 1?)? p (x l j Y 0?)g (dx l )j P ( l 2 S + j Y 1?)? P ( l 2 S + j Y 0?) i=l (1?!(Y i )); where S + = fx l 2 j p (x l j Y? 1 )? p (x l j Y? 0 ) > 0g is chose as i Lemma 4.2, ad the last iequality follows from 4.5. The secod iequality is derived by a similar argumet, by coditioig o k istead of l. I the ext Lemma f k g deotes ay ihomogeous Markov chai, as i Lemma 4.3. Lemma 5.2 Let the setup be as i Lemma 4.3. Let 2 ad let Q be the measure o A A deed by, for A; B 2 A. The for all C 2 A A, Q(A B) = P ( 0 2 A)P ( 2 B); jp (( 0 ; ) 2 C)? Q(C)j Y k=1 (1? k ( )): Proof Let C x0 = fx 2 j (x 0 ; x ) 2 Cg; the jp (( 0 ; ) 2 C)? Q(C)j = j fp ( 2 C x0 j 0 = x 0 )? P ( 2 C x0 )g P 0 (dx 0 )j Here the last iequality follows from Lemma 4.3 sice Y k=1 (1? k ( )): jp ( 2 A j 0 = )? P ( 2 A)j = j fp ( 2 A j 0 = )? P ( 2 A j 0 = )g P 0 (d) = sup P ( 2 A j 0 = )? if P ( 2 A j 0 = ) 2 2 Y k=1 (1? k ( )): Lemma 5.3 Let?m? k l 0. Let Q j ;? be the measure o A A deed by Q j ;? (A B) = P ( k 2 A j Y j?)p ( l 2 B j Y j?)
14 J. L. Jese ad N. V. Peterse for j = 0; 1 ad A; B 2 A. The for all 2 B 0, for C 2 A A ad for j = 0; 1, jp (( k ; l ) 2 C j Y j?)? Q j ;?(C)j jq 1 ;?(C)? Q 0 ;?(C)j 2 jq j ;?(C)? Q j ;?m(c)j 2 i=l ky i=? l?1 Y i=k (1?!(Y i )); (1?!(Y i )); (1?!(Y i )): Proof The rst iequality follows from Lemma 5.2 ad 4.1. To prove the secod expressio we will let C y = fx 2 j (x; y) 2 Cg, C 0 x = fy 2 j (x; y) 2 Cg ad proceed as follows, jq 1 ;? (C)? Q0 ;? (C)j = j j C i=k C + j + fp (x k j Y 1? )p (x l j Y 1? )? p (x k j Y 0? )p (x l j Y 0? )g (dx k)(dx l )j fp (x k j Y 1? )? p (x k j Y 0? )g p (x l j Y 1? )(dx k)(dx l )j C fp (x l j Y 1?)? p (x l j Y 0?)g p (x k j Y 0?) (dx l )(dx k )j jp ( k 2 C xl j Y 1?)? P ( k 2 C xl j Y 0?)j p (x l jy 1?) (dx l ) + jp ( l 2 C 0 x k j Y 1?)? P ( l 2 C 0 x k j Y 0?)j p (x k jy 0?) (dx k ) (1?!(Y i )) + i=l (1?!(Y i )) 2 where the third iequality is give by Lemma 4.5. The third expressio is proved as the secod. i=l (1?!(Y i )); Havig established these iequalities we are ready to prove the law of large umbers for the observed iformatio. Usig A5(i) to iterchage itegratio ad dieretiatio we d D 2 log p (Y 1 j Y 0?) = D 2 log p (Y 1?)? D 2 log p (Y 0?) (12) = E (D 2 log p ( 1? ; Y 1? ) j Y 1? )? E (D 2 log p ( 1? ; Y 0? ) j Y 0? ) + var (D log p ( 1?; Y 1?) j Y 1?)? var (D log p ( 1?; Y 0?) j Y 0?) Dee for otatioal reasos ;k = ( k ; k+1 ) + (Y k j k ) ad _ ;k = D ( k ; k+1 ) + D (Y k j k ):
Asymptotics i state space models 15 Isertig expressio (1) i (12) we get, D 2 log p (Y 1 j Y?) 0 (13) = E (D (? ) j Y 1 )?? E (D (? ) j Y 0 ) +? E (D (Y 1 j 1 ) j Y 1 )? + 0 k=? fe ( _ ;k j Y 1?)? E ( _ ;k j Y 0?)g + var ( (Y 1 j 1 ) j Y 1?) + var ( (? ) j Y 1?)? var ( (? ) j Y 0?) + + 2 + 2 0 0 k=? l=? 0 k=? 0 k=? cov ( ;k ; ;l j Y 1?)? cov ( ;k ; ;l j Y 0?) o cov ( (? ); ;k j Y 1? )? cov ( (? ); ;k j Y 0? )o cov ( (Y 1 j 1 ); ;k j Y 1?) + 2cov ( (Y 1 j 1 ); (? ) j Y 1?): We the have the followig covergece result. Lemma 5.4 As m;! 1, sup jd 2 log p (Y 1 j Y?) 0? D 2 log p (Y 1 j Y?m)j 0 2B 0 where jjjj 1 deotes the L 1 (P 0 )-orm. 1! 0; This Lemma states that fd 2 log p (Y 1 j Y?)g 0 is a uiform Cauchy sequece i L 1 (P 0 ). This is importat because it proves the existece of a limit i L 1 (P 0 ) of D 2 log p (Y 1 j Y 1 ) as?! 1 for ay 2 B 0, ad ot oly = 0. I the proof we will eed the followig Lemma. Lemma 5.5 Let?m? k l 0 ad let ;k be deed as above. The there exists a 2 [0; 1) such that the followig iequalities hold for j = 0; 1, 1 sup jcov ( ;k ; ;l j Y 1 )? cov? ( ;k ; ;l j Y 0 )j? c?l ; (14) 2B 0 sup jcov ( ;k ; ;l j Y?) j? cov ( ;k ; ;l j Y?m)j j 2B 0 c k+ ; (15) 1 sup c l?k : (16) 2B 0 jcov ( ;k ; ;l j Y j?)j Above ;i may be replaced by ( i ) or (Y i j i ) for i = k; l. 1 Proof Recall that ;k = ( k ; k+1 ) + (Y k j k ). Thus the covariace of ;k ad ;l splits ito the sum of four covariace terms ivolvig ad. We will
16 J. L. Jese ad N. V. Peterse show that cov ( (Y k j k ); (Y l j l ) j Y ) satises the claimed iequalities. The three remaiig terms are similar. To show the rst iequality we will cosider the expressio sup je f (Y k j k ) (Y l j l ) j Y 1 g?? E f (Y k j k ) (Y l j l ) j Y 0 gj? 2B 0 = sup j (Y k j x k ) (Y l j x l )fp (x k ; x l j Y?) 1? p (x k ; x l j Y?)g 0 (dx k )(dx l )j 2B 0 2 2 (Y k ) (Y l ) sup jp (( k ; l ) 2 S + j Y?) 1? P (( k ; l ) 2 S + j Y?)j 0 2B 0 2 (Y k ) (Y l ) i=l (1?!(Y i )): Here (Y k ) = sup 2B0 sup x2 j (Y k jx)j as deed i assumptio A4, ad the iequalities follows from Lemma 4.2 ad 5.1, respectively. The L 1 (P 0 )-orm of such a term is thus less tha 2E 0 (Y k ) (Y l ) i=l = 2E 0 E 0 (Y k ) (Y l ) (1?!(Y i )) i=l! (1?!(Y i )) 0 k 2E 0 E 0 ( (Y k )j k )E 0 ( (Y l )j l ) 2E 0 ( (Y 1 ) 2 )?l = c?l ; i=l+1!! E 0 (1?!(Y i )j i ) where the rst iequality follows by deitio of the state space model, ad where is give by = sup x2 E 0 (1?!(Y i )j i = x) = sup x2 Y 1?!! ( )2 g 0 (yjx) (dy) < 1; M(y) by assumptio A3. Assumptio A4 assures that the costat c above is ite. The expressio sup je f (Y k j k ) j Y 1? ge f (Y l j l ) j Y 1 g?? 2B 0 E f (Y k j k ) j Y 0?gE f (Y l j l ) j Y 0?gj ca be domiated by the same techique, usig the secod expressio i Lemma 5.3. Hece (14) is proved. 1
Asymptotics i state space models 17 The secod iequality (15) is proved as (14) usig the secod expressio i Lemma 5.1 ad the third expressio i Lemma 5.3, respectively. As for (16) we have sup jcov ( (Y k j k ); (Y l j l ) j Y?)j j 2B 0 = sup 2B 0 j 2 (Y k j x k ) (Y l j x l ) 2 (Y k ) l?1 (Y l ) (1?!(Y i )); fp (x k ; x l j Y j?)? p (x k j Y j?)p (x l j Y j?)g (dx k )(dx l )j Y i=k by the rst expressio i Lemma 5.3. The claimed domiatio of the L 1 (P 0 )-orm of this term is proved as above. Proof (Lemma 5.4.) Cosiderig the expressio for D 2 log p (Y 1 j Y?) 0 i (13) we will show that the term sup 2B 0 0 0 k=?m l=?m? cov ( ;k ; ;l j Y 1?m )? cov ( ;k ; ;l j Y 0?m )o 0 k=? 0 l=? cov ( ;k ; ;l j Y 1?)? cov ( ;k ; ;l j Y 0?) o 1! 0 (17) as ; m! 1. The remaiig terms i (13) ca be treated with similar argumets. Suppose m >. By symmetry of k ad l i the sum i (17) it suces to cosider the sum over the regio where k l. This regio ca be further divided ito 5 subregios, D 1 = f(k; l) 2 2 j? [=2] k 0; k l 0g; D 2 = f(k; l) 2 2 j? k?[=2]; [k=2] l 0g; D 3 = f(k; l) 2 2 j? m k?; [k=2] l 0g; D 4 = f(k; l) 2 2 j? k?[=2]; k l [k=2]g; D 5 = f(k; l) 2 2 j? m k?; k l [k=2]g: We will show that the sum over each of these regios teds to zero i L 1 (P 0 ) as ; m! 1, hece provig (17). Usig (15) we d that (k;l)2d 1 sup jfcov ( ;k ; ;l j Y?m) 1? cov ( ;k ; ;l j Y?m)g 0 2B 0? fcov ( ;k ; ;l j Y 1? )? cov ( ;k ; ;l j Y 0? )gj 1 c 0 0 k=?[=2] l=k Usig (16) we d that the correspodig sums over D 2 ad D 3 are less tha?[=2] c 0 k=? l=[k=2] l?k ad c? 0 k=?m l=[k=2] l?k ; k+
18 J. L. Jese ad N. V. Peterse respectively, ad by (14) the sums over D 4 ad D 5 are domiated by c?[=2] k=? [k=2] l=k?l ad c? [k=2] k=?m l=k respectively. Sice 0 < 1 these sums all ted to zero as ; m! 1 ad the proof is complete.?l ; Lemma 5.6 The map 7! D 2 log p (Y 1 jy 0? ) from B 0 to L 1 (P 0 ) is cotiuous. Proof Let f m g B 0 be a sequece such that m! as m! 1. We will show that E 0 fjd 2 log p m (Y 1 j Y 0?)? D 2 log p (Y 1 j Y 0?)jg! 0; as m! 1: Cosiderig the expressio i (13) we must show that terms like, for istace, E 0 h jem f m (Y k j k ) m (Y l j l ) j Y 1? g? E f (Y k j k ) (Y l j l ) j Y 1? gji ted to zero as m! 1. The itegrad ca be evaluated as je m f m (Y k j k ) m (Y l j l ) j Y 1?g? E f (Y k j k ) (Y l j l ) j Y 1?gj j + j 2 m (Y k jx k ) m (Y l jx l )fp m (x k ; x l j Y 1? )? p (x k ; x l j Y 1? )g (dx k)(dx l )j 2 f m (Y k jx k ) m (Y l jx l )? (Y k jx k ) (Y l jx l )gp (x k ; x l j Y 1?) (dx k )(dx l )j: The rst term is less tha (Y k ) (Y l ) = (Y k ) (Y l ) p m (Y 1?) 2 jp m (x k ; x l j Y 1?)? p (x k ; x l j Y 1?)j (dx k )(dx l ) (18) 2 jp m (x k ; x l ; Y 1? )? p (x k ; x l j Y 1? )p m (Y 1? )j (dx k)(dx l ): The itegral teds to zero as m! 1 as ca be see by cosiderig the simultaeous desity p m (x k ; x l ; Y 1?) = m (x? )g m (Y? jx? ) 1Y i=?+1 f m (x i?1 ; x i )g m (Y i jx i )g Sice the itegrad here is cotiuous ad ca be domiated by M +2 1Y i=? 1Y i=? i6=k;l (19) (dx i ): (20) h Yi (x i ) 2 L 1 ( ); (21) by assumptio A1 ad A5, we have from Lebesgue's domiated covergece theorem that p m (x k ; x l ; Y 1 )!? p (x k ; x l ; Y 1?) as m! 1:
Asymptotics i state space models 19 Likewise p m (Y 1 )!? p (Y 1?) as m! 1, ad hece the itegrad i (19) teds to zero. By (21) the itegrad ca be domiated i L 1 ( 2 ), ad therefore (19) teds to zero. Sice the expressio i (18) is less tha (Y k ) (Y l ) fp m (x k ; x l jy?) 1 + p (x k ; x l jy?)g 1 (dx k )(dx l ) 2 = 2 (Y k ) (Y l ); (22) it is domiated i L 1 (P 0 ) ad hece teds to zero i L 1 (P 0 ) as m! 1: The secod term ca be domiated similarly ad teds to zero P 0 -almost surely, ad therefore also i L 1 (P 0 ), by the cotiuity of. Lemma 5.4 ad 5.6 show that fd 2 log p (Y 1 jy 0? )g 2N is a uiform Cauchy sequece of cotiuous fuctios i L 1 (P 0 ), which proves Lemma 10 of Bickel et al. The al Lemma states a usual property of the Fisher iformatio. With this result, the remaiig part of the proof of Theorem 3.2 is ow idetical to the proof of Lemma 2 i Bickel et al. (p. 1633) Lemma 5.7 For ay, E 0 fd 2 log p 0 (Y 1 jy 0?)g =?E 0 f[d log p 0 (Y 1 jy 0?)] 2 g: Proof By (3) ad (12) we have (D log p 0 (Y 1 j Y 0?)) 2 + D 2 log p 0 (Y 1 j Y 0?) = 2 E 0 (D log p 0 ( 1?; Y 0?) j Y 0?) 2? E 0 D log p0 ( 1? ; Y 1? ) j Y 1? E0 D log p0 ( 1? ; Y 0? ) j Y 0? + E 0 ((D log p 0 ( 1?; Y 1?)) 2 j Y 1?)? E 0 ((D log p 0 ( 1?; Y 0?)) 2 j Y 0?) + E 0 (D 2 log p 0 ( 1? ; Y 1? ) j Y 1? )? E 0(D 2 log p 0 ( 1? ; Y 0? ) j Y 0? ):! (23) The rst term has zero mea. This follows by otig from (1) that D log p 0 ( 1? ; Y 1? ) = D log p 0( 1? ; Y 0? ) + 0(Y 1 j 1 ); (24)
20 J. L. Jese ad N. V. Peterse thus E 0 h E0 fd log p 0 ( 1?; Y 1?) j Y 1?gE 0 fd log p 0 ( 1?; Y 0?) j Y 0?g i = E 0 D log p0 ( 1? ; Y 1? )E 0fD log p 0 ( 1? ; Y 0 ) j? Y 0 go? o = E 0 D log p 0 (?; 1 Y?)E 0 0 (D log p 0 (?; 1 Y?) 0 j Y?) 0 + E 0 0 (Y 1 j 1 )E 0 (D log p 0 ( 1?; Y 0?) j Y 0?) o = E 0 D log p0 ( 1?; Y 0?)E 0 (D log p 0 ( 1?; Y 0?) j Y 0?) o + E 0 E 0 h 0 (Y 1 j 1 )j 1 i E 0 h E 0 (D log p 0 ( 1?; Y 0?) j Y 0?) j 1 io = E 0 D log p0 ( 1?; Y 0?)E 0 (D log p 0 ( 1?; Y 0?) j Y 0?) o ; where the third equality follows from the coditioal idepedece of Y 0? ad Y 1 give 1, ad the last equality from the fact that E 0 ( 0 (Y 1 j 1 ) j 1 ) = 0 by A5(ii). The mea of the rst term i (23) is the 2E 0 E0 (D log p 0 ( 1?; Y 0?) j Y 0?) h E0 (D log p 0 ( 1?; Y 0?) j Y 0?)? D log p 0 ( 1?; Y 0?) io = 0: By (24) the mea of the sum of the two last terms is give by E 0 D2 log g 0 (Y 1 j 1 ) o + E 0 (D log g0 (Y 1 j 1 )) 2 o + 2E 0 0 (Y 1 j 1 )D log p 0 ( 1?; Y 0?) o : The last term is zero, which is see by coditioig o 1 ad usig the argumet from above. The sum of the two rst terms is zero by assumptio A5(ii), which completes the proof. Refereces Bickel, P. J., Ritov, Y. & Ryde, T. (1998), `Asymptotic ormality of the maximumlikelihood estimator for geeral hidde Markov models', A. Statist. 26(4), 1614{ 1635. Doob, J. L. (1953), Stochastic Processes, Joh Wiley & Sos, Ic. Durbi, J. & Koopma, S. J. (1997), `Mote Carlo maximum likelihood estimatio for o-gaussia state space models', Biometrika 84(3), 669{684. Fruhwirth-Schatter, S. (1994), `Applied state space modellig of o-gaussia time series usig itegratio-based Kalma lterig', Stat. Comp. 4, 259{269. Homa-Jrgese, J. (1994), Probability with a View toward Statistics, Vol. 1, Chapma & Hall.
Asymptotics i state space models 21 Jese, J. L. (1986), Nogle asymptotiske resultater. Uiversity of Aarhus (I Daish). Kitagawa, G. & Gersch, W. (1996), Smoothess Priors Aalysis of Time Series, Spriger-Verlag, New York. Leroux, B. G. (1992), `Maximum-likelihood estimatio for hidde Markov models', Stoch. Proc. Appl. 40, 127{143. Shephard, N. & Pitt, M. K. (1997), `Likelihood aalysis of o-gaussia measuremet time series', Biometrika 84(3), 653{667. Sweetig, T. (1980), `Uiform asymptotic ormality of the maximum likelihood estimator', A. Statist. 8(6), 1375{1381. West, M. & Harriso, J. (1989), Bayesia Forecastig ad Dyamic Models, Spriger- Verlag, New York. Dept. of Theoretical Statistics Uiversity of Aarhus Ny Mukegade DK-8000 Aarhus C, Demark E-mail: jlj@imf.au.dk vaever@imf.au.dk