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

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. Unlike 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. The convergence of the maximum likelihood estimator is illustrated on simulated data.