ASYMPTOTIC PROPERTIES OF THE MLE IN HIDDEN MARKOV MODELS

S
i;j q = P [Xn+1 = j j Xn = i] ; and initial probability distribution p0 = (pi0 ) independent of 2 , and possibly di erent of the true initial probability distribution p = (pi ) of X0 , i.e. for any i 2 S 4 pi = P [X0 = i] 6= P[X0 = i] = pi : 0
; F ; P), with values in the nite set S = f1; ; N g and Rd respectively. On the space (
; F ) we consider a family (P ; 2 ) of probability measures, with compact subset of Rp , such that under P : The unobserved state sequence fXn ; n 0g is a Markov chain with transition probability matrix i;j the primitive matrix Q = (q ), i.e. for any i; j 2
1 Introduction
In this paper, we consider the problem of identi cation of a partially observed nite{state Markov chain, based on observations in Rd. In a rst part, we investigate the consistency of the maximum likelihood estimator (MLE) as the number of observations increases to in nity. In the case of observations in a nite set, the problem has already been considered by Petrie [7], see also Le Gland and Mevel [3] for a study of the recursive MLE, and our contribution is to generalize his work to continuous observations, using results on the approximate prediction lter obtained in Le Gland and Mevel [4]. The consistency of the MLE has already been studied by Leroux [6] under stationarity of the Markov chain. In Rydn [9], this work has been e extended to Markov modulated Poisson process. In a second part, we are interested in conditions for the asymptotic normality of the MLE. The property has been obtained by Rydn [8] for the maximum split data e likelihood estimator (MSDLE) de ned therein.
Remark that all the tools used in this paper can be used to prove similar results about some others estimators. For example, we prove the consistency and the asymptotic normality of the conditionnal least squares estimator in LeGland and Mevel [5]. The statistical model is as follows. Let fXn ; n 0g and fYn ; n 0g be two sequences, de ned on the probability space (
This
The observations fYn ; n 0g are mutually independent given the sequence of states of the Markov chain, i.e.
P [Yn 2 dyn; ; Y0 2 dy0 j
j Xn = in ; ; X0 = i0 ] =
(y) =
Example 1.1 [Conditionally Gaussian observations] Assume that for any 2 , the observations are of
the form
Yn = h (Xn ) + Vn ; for all n 0, where, under P , fVn ; n 0g is a Gaus-
work was partially supported by the Commission of the European Communities, under the SCIENCE project System Identi cation, project number SC1*{CT92{0779, and under the HCM project Statistical Inference for Stochastic Processes, project number CHRX{CT92{0078, and by the Army Research Oce, under grant DAAH04{95{1{0164.
=
n Y k=0
P [Yk 2 dyk j Xk = ik ] :
For any n 0, and for any i 2 S , the conditional probability distribution of the observation Yn given that (Xn = i) is absolutely continuous with respect to a unique positive and { nite measure on Rd, i.e. for any i 2 S , with a {a.e. positive dn = i] = bi (y) (dy) ;
For any y 2 Rd, let b (y) = [ b1 (y); ; bN (y) ] ;
B (y) = diag[b1 (y); ; bN (y)] :
Assumption B' : For any y 2 Rd, the mapping ! b (y) is three{times di erentiable. For any y 2 Rd, and any 2 , de ne s max k ;;k =1;;p j@k0 ;;ks bi (y)j max i2S 0 s s
ASYMPTOTIC PROPERTIES OF THE MLE IN HIDDEN MARKOV MODELS
F. LeGland, L. Mevel IRISA / INRIA and IRMAR Campus de Beaulieu 35042 RENNES Cdex, France e fax : +33 2 99 84 71 71 | e-mail : flegland,lmevelg@irisa.fr Keywords : HMM, estimation, stochastic.
1 2 jy
hi j2
;
Assumption A : For the true value 2 , the tran-
ij sition probability matrix Q = (q ) is a positive matrix. This implies that Q is an irreducible and aperiodic matrix. Moreover, there exists some ", possibly i;j unknown, such that q ", for all i; j 2 S , and we de ne " = f 2 ; d(; @ ) "g : Remark 1.2 For the proof of the consistency, the assumption of positivity of the transition probability matrix Q can be relaxed to primitivity.
bi (y) = (2) d=2 exp

for any i 2 S . Finally, we add the following regularity assumptions about the transition probability matrix, and the observation densities :
sian white noise sequence, with identity covariance matrix. The mapping h from S to Rd is equivalently de ned as h = (hi ) where hi 2 Rd for all i 2 S . In this case, the mutual independence condition is satis ed, and