  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 nonparametric maximum likelihood
estimation theory. First, we show that the asymptotic contrast, the
Kullback--Leibler divergence of the hidden Markov model, identifies the true
parameter vector nonparametrically as well. Second, for classes of
state-dependent densities which are arbitrary mixtures of a parametric family,
we show consistency of the nonparametric maximum likelihood estimator. Here,
identification of the mixing distributions need not be assumed. Numerical
properties of the estimates as well as of nonparametric goodness of fit tests
are investigated in a simulation study.
