. The Bayesian Information Criterion (BIC) estimates the order of a Markov chain (with finite alphabet A) from observation of a sample path x 1 ; x 2 ; : : : ; x n , as that value k = k that minimizes the sum of the negative logarithm of the k-th order maximum likelihood and the penalty term jAj k (jAj\Gamma1) 2 log n: We show that k equals the correct order of the chain, eventually almost surely as n ! 1, thereby strengthening earlier consistency results that assumed an apriori bound on the order. A key tool is a strong ratio-typicality result for Markov sample paths. We also show that the Bayesian estimator or minimum description length estimator, of which the BIC estimator is an approximation, fails to be consistent for the uniformly distributed i.i.d. process. AMS 1991 subject classification: Primary 62F12, 62M05; Secondary 62F13, 60J10 Key words and phrases: Bayesian Information Criterion, order estimation, ratiotypicality, Markov chains. 1 Supported in part by a joint N...

The concept of context tree, usually defined for finite memory processes, is extended to arbitrary stationary ergodic processes (with finite alphabet). These context trees are not necessarily complete, and may be of infinite depth. The familiar BIC and MDL principles are shown to provide strongly consistent estimators of the context tree, via optimization of a criterion for hypothetical context trees of finite depth, allowed to grow with the sample size n as o(log n). Algorithms are provided to compute these estimators in O(n) time, and to compute them on-line for all i ≤ n in o(n log n) time.

We consider the estimation of the order, i.e., the number of hidden states, of a special class of discrete-time finite-alphabet hidden Markov sources. This class can be characterized in terms of equivalent renewal processes. No a priori bound is assumed on the maximum permissible order. An order estimator based on renewal types is constructed, and is shown to be strongly consistent by computing the precise asymptotics of the probability of estimation error. The probability of underestimation of the true order decays exponentially in the number of observations while the probability of overestimation goes to zero sufficiently fast. It is further shown that this estimator has the best possible error exponent in a large class of estimators. Our results are also valid for the general class of binary independent-renewal processes with finite mean renewal times.

Abstract. We present two new methods for estimating the order (memory depth) of a finite alphabet Markov chain from observation of a sample path. One method is based on entropy estimation via recurrence times of patterns, and the other relies on a comparison of empirical conditional probabilities. The key to both methods is a qualitative change that occurs when a parameter (a candidate for the order) passes the true order. We also present extensions to order estimation for Markov random fields.

For Markov random fields on Z d with finite state space, we address the statistical estimation of the basic neighborhood, the smallest region that determines the conditional distribution at a site on the condition that the values at all other sites are given. A modification of the Bayesian Information Criterion, replacing likelihood by pseudo-likelihood, is proved to provide strongly consistent estimation from observing a realization of the field on increasing finite regions: the estimated basic neighborhood equals the true one eventually almost surely, not assuming any prior bound on the size of the latter. Stationarity of the Markov field is not required, and phase transition does not affect the results. 1. Introduction. In this paper Markov

. We announce two results on the problem of estimating the order of a Markov chain from observation of a sample path. First is that the Bayesian Information Criterion (BIC) leads to an almost surely consistent estimator. Second is that the Bayesian minimum description length estimator, of which the BIC estimator is an approximation, fails to be consistent for the uniformly distributed i.i.d. process. A key tool is a strong ratio-typicality result for empirical k-block distributions. Complete proofs are given in the authors' article to appear in The Annals of Statistics. 1. Introduction Let M k denote the class of Markov chains of order at most k, with values drawn from a finite set A, and let M = S 1 k=0 M k . An important problem is to estimate the order of a Markov chain from observation of a finite sample path. A popular method is the so-called Bayesian Information Criterion (BIC), first introduced by Schwarz, [12], which gives the estimator defined by k BIC = k BIC (x n 1 ) ...

In this paper we derive the consistency of the penalized likelihood method for the number state of the hidden Markov chain in autoregressive models with Markov regimen. Using a SAEM type algorithm to estimate the models parameters. We test the null hypothesis of hidden Markov Model against an autoregressive process with Markov regime.