Jean Diebolt

Bayesian estimation of hidden Markov chains: a stochastic implementation

Hidden Markov models lead to intricate computational problems when considered directly. In this paper, we propose an approximation method based on Gibbs sampling which allows an effective derivation of Bayes estimators for these models.

A stochastic approximation type EM algorithm for the mixture problem

The EM algorithm is a widely applicable approach for computing maximum likelihood estimates for incomplete data. We present a stochastic approximation type EM algorithm: SAEM. This algorithm is an adaptation of the stochastic EM algorithm (SEM) that we have previously developed. Like SEM, SAEM overcomes most of the well-known limitations of EM. Moreover, SAEM performs better for small samples. Furthermore, SAEM appears to be more tractable than SEM, since it provides almost sure convergence, while SEM provides convergence in distribution. Here, we restrict attention on the mixture problem. We state a theorem which asserts that each SAEM sequence converges a.s. to a local maximizer of the likelihood function. We close this paper with a comparative study, based on numerical simulations, of these three algorithms.

Asymptotic Properties of a Stochastic EM Algorithm for Estimating Mixing Proportions

The purpose of this paper is to study the asymptotic behavior of the Stochastic EM algorithm (SEM) in a simple particular case within the mixture context. We consider the estimation of the mixing proportion p: of a two-component mixture of densities assumed to be known. We establish that as the sample size N tends to infinity, the stationary distribution of the ergodic Markov chain generated by SEM converges to a Gaussian distribution whose mean is the consistent maximum likelihood estimate of p:. The asymptotic variance is proportional to N -1

A nonparametric test for the regression function: Asymptotic theory

Abstract This paper is devoted to the study of the asymptotic behavior of a family of nonparametric tests for the regression function in a nonlinear context, when the observations are i.i.d. These nonparametric tests are all constructed as functionals of a basic process. We first determine the rate of convergence to zero of the distance between this basic process and Wiener type approximations. This involves establishing uniform rates of convergence for a certain sequence of processes to their limit. These processes are hybrids of the empirical and partial-sum processes, and are of some interest themselves. Then, we examine the asymptotic behavior of the power of two particular tests within the family under consideration. Both tests have been selected as the most natural ones in some sense. Contiguous alternatives are also briefly examined for these tests.

Testing the functions defining a nonlinear autoregressive time series

We first establish the consistency of regressogram type estimators of the functions T and U based on he observation of the processXn+1=T(Xn)+U(Xn)[var epsilon]n+1, then nonparametric goodness-of-fit tests for the functions T and U are introduced and discussed. These nonparametric tests constitute the main contribution of this article.

Control by the Central Limit Theorem

Distinctions between single chain and parallel chain control methods have already been discussed in Chapter 2. However, as Brooks and Roberts (1998) point out, other characteristics must be taken into account for evaluating control methods. An important criterion is the programming investment: diagnostics requiring problem-specific computer codes for their implementation (e.g., requiring knowledge of the transition kernel of the Markov chain) are far less usable for the end user than diagnostics solely based upon the outputs from the sampler, which can use available generic codes. Another criterion is interpretability, in the sense that a diagnostic should preferably require no interpretation or experience from the user.

Estimates of the tail of the stationary density function of certain nonlinear autoregressive processes

We consider the Markov chainXn+1=T(Xn)+εn, where {εn;n⩾1} is a ℜd-valued random sequence of independent identically distributed random variables, and the functionT: ℜd→ℜd is measurable and satisfies a suitable growth condition. Under certain conditions involvingT and the probability distribution of εn, we show that this Markov chain is ergodic. Moreover, we obtain sharp upper bounds for the tail of the corresponding stationary probability density function. In our proofs, we make use of the Leray-Schauder fixed-point theorem.