  In the paper, we introduce the maximum entropy estimator based on
2-dimensional empirical distribution of the observation sequence of hidden
Markov model , when the sample size is big: in that case computing the maximum
likelihood estimator is too consuming in time by the classical Baum-Welch EM
algorithm. We prove the consistency and the asymptotic normality of the maximum
entropy estimator in a quite general framework, where the asymptotic covariance
matrix is explicitly estimated in terms of the 2-dimensional Fisher
information. To complement it, the 2-dimensional relative entropy is skillfully
used to study the hypotheses testing problem. Furthermore, we propose
2-dimensional maximum entropy algorithm for finding the maximum entropy
estimator, which works for very large observation dataset and large hidden
states set. Some numerical examples are furnished and commented for
illustrating our theoretical results.
