Asymptotic properties of the maximum likelihood estimation in misspecified hidden Markov models

TL;DR

This paper investigates the asymptotic behavior and consistency of maximum likelihood estimators in misspecified hidden Markov models, where the true data-generating process may not belong to the model family.

Contribution

It provides theoretical results on the consistency of MLE in misspecified HMMs under mild conditions, extending understanding beyond correctly specified models.

Findings

MLE is consistent under mild assumptions despite model misspecification

The paper establishes asymptotic properties of the estimator in complex settings

Results apply to a broad class of stationary processes and HMMs

Abstract

Let $(Y_k)_{k\in \mathbb{Z}}$ be a stationary sequence on a probability space $(\Omega,\mathcal{A},\mathbb{P})$ taking values in a standard Borel space $\mathsf{Y}$. Consider the associated maximum likelihood estimator with respect to a parametrized family of hidden Markov models such that the law of the observations $(Y_k)_{k\in \mathbb{Z}}$ is not assumed to be described by any of the hidden Markov models of this family. In this paper we investigate the consistency of this estimator in such misspecified models under mild assumptions.