Consistency of the maximum likelihood estimator in hidden Markov models with trends

TL;DR

This paper proves that in hidden Markov models with polynomial trends, the maximum likelihood estimator consistently recovers both the hidden states and the trends, despite the non-homogeneity introduced by the trends.

Contribution

It establishes the strong consistency of the maximum likelihood estimator in non-homogeneous hidden Markov models with polynomial trends, a novel extension beyond standard models.

Findings

MLE recovers true trends with supremum norm convergence

Simulation confirms the estimator's numerical stability

Trends can be accurately estimated alongside hidden states

Abstract

A hidden Markov model with trends is a hidden Markov model whose emission distributions are translated by a trend that depends on the current hidden state and on the current time. Contrary to standard hidden Markov models, such processes are not homogeneous and cannot be made homogeneous by a simple de-trending step. We show that when the trends are polynomial, the maximum likelihood estimator is able to recover the trends together with the other parameters and is strongly consistent. More precisely, the supremum norm of the difference between the true trends and the estimated ones tends to zero. Numerical properties of the maximum likelihood estimator are assessed by a simulation study.