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.

翻译：隐藏的Markov 模式具有趋势,是一种隐蔽的Markov 模式,其排放分布由取决于当前隐藏状态和当前时间的趋势所翻译。与标准的Markov 模式相反,这种过程不均匀,不能通过简单的脱调步骤而实现同质。我们显示,当趋势是多数值时,最大可能性的估测器能够与其他参数一起恢复趋势,并且非常一致。更确切地说,真实趋势与估计趋势之间差异的超值标准往往为零。最高概率估测器的数值由模拟研究来评估。