Consistent Estimation of the Order of Hidden Markov Chains

The structural parameters of many statistical models can be estimated maximizing a penalized version of the likelihood function. We use this idea to construct strongly consistent estimators of the order of Hidden Markov Chain models. The specification of the penalty term requires precise information on the rate of growth of the maximized likelihood ratio. We find an upper bound to the rate using results from Information Theory. We give sufficient conditions on the penalty term to avoid overestination and underestimation of the order. Examples of penalty terms that generate strongly consistent estimators are also given.