Nonparametric identification and maximum likelihood estimation for finite-state hidden Markov models are investigated. We obtain identification of the parameters as well as the order of the Markov chain if the transition probability matrices have full-rank and are ergodic, and if the state-dependent distributions are all distinct, but not necessarily linearly independent. Based on this identification result, we develop a nonparametric maximum likelihood estimation theory. First, we show that the asymptotic contrast, the Kullback-Leibler divergence of the hidden Markov model, also identifies the true parameter vector nonparametrically. Second, for classes of state-dependent densities which are arbitrary mixtures of a parametric family, we establish the consistency of the nonparametric maximum likelihood estimator. Here, identification of the mixing distributions need not be assumed. Numerical properties of the estimates and of nonparametric goodness of fit tests are investigated in a simulation study.

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We review the theory on semiparametric hidden Markov models (HMMs), that is, HMMs for which the state-dependent distributions are not fully parametrically, but rather semi-or nonparametrically specified. We start by reviewing identifiability in such models, where by exploiting the dependence much stronger results can be achieved than for independent finite mixtures. We also discuss estimation, in particular in an algorithmic fashion by using appropriate versions or modifications of the Baum-Welch (or EM) algorithm. We present some simulation results and give an application to modeling bivariate financial time series, where we compare parametric with nonparametric fits for the state-dependent distributions as well as the resulting state-decoding.Conflict of interest: The authors have declared no conflicts of interest for this article. Recently, there has been some interest in a semi-or even fully nonparametric specification of the state-dependent distributions, cf. Refs 12-16 for some applications of such models. We shall call the resulting HMMs as semiparametric HMMs. Semiparametric modeling may be of interest for the following reasons:418