Least squares type estimation of the transition density of a particular hidden Markov chain

TL;DR

This paper introduces a novel least squares-based estimator for the transition density of a hidden Markov chain with known noise distribution, achieving adaptive, rate-optimal convergence without prior regularity knowledge.

Contribution

It proposes an original contrast minimization approach combined with model selection for adaptive transition density estimation in hidden Markov models.

Findings

Achieves optimal convergence rates for the estimator.

Provides uniform risk bounds over Besov spaces.

Demonstrates effectiveness through simulations.

Abstract

In this paper, we study the following model of hidden Markov chain: $Y_i=X_i+\epsilon_i$, $i=1,...,n+1$ with $(X_i)$ a real-valued stationary Markov chain and $(\epsilon_i)_{1\leq i\leq n+1}$ a noise having a known distribution and independent of the sequence $(X_i)$. We present an estimator of the transition density obtained by minimization of an original contrast that takes advantage of the regressive aspect of the problem. It is selected among a collection of projection estimators with a model selection method. The $L^2$-risk and its rate of convergence are evaluated for ordinary smooth noise and some simulations illustrate the method. We obtain uniform risk bounds over classes of Besov balls. In addition our estimation procedure requires no prior knowledge of the regularity of the true transition. Finally, our estimator permits to avoid the drawbacks of quotient estimators.