Selecting the number of regimes in Hidden Markov models is an important problem. There are many criteria that are used to select this number, such as Akaike information criterion (AIC), Bayesian information criterion (BIC), integrated completed likelihood (ICL), deviance information criterion (DIC), and Watanabe-Akaike information criterion (WAIC), to name a few. In this article, we introduced goodness-of-fit tests for general Hidden Markov models with covariates, where the distribution of the observations is arbitrary, i.e., continuous, discrete, or a mixture of both. Then, a selection procedure is proposed based on this goodness-of-fit test. The main aim of this article is to compare the classical information criterion with the new criterion, when the outcome is either continuous, discrete or zero-inflated. Numerical experiments assess the finite sample performance of the goodness-of-fit tests, and comparisons between the different criteria are made.

翻译：选择隐马尔可夫模型中的状态数是一个重要问题。有许多准则用于选择该数量，例如Akaike信息准则（AIC）、贝叶斯信息准则（BIC）、综合完全似然准则（ICL）、偏差信息准则（DIC）和Watanabe-Akaike信息准则（WAIC）等。在本文中，我们引入了带协变量的一般隐马尔可夫模型的拟合优度检验，其中观测值的分布是任意的，即连续型、离散型或两者的混合。然后，基于该拟合优度检验提出了一种选择程序。本文的主要目的是比较经典信息准则与新准则在结果为连续型、离散型或零膨胀型时的表现。数值实验评估了拟合优度检验的有限样本性能，并对不同准则进行了比较。