Consistent Estimation of the Order for Markov and 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 for Markov Chains and Hidden Markov Chain models. The specification of the penalty term requires a precise information on the rate of growth of the maximized likelihood ratio. For Markov chain models we determine the rate using the Law of the Iterated Logarithm. For Hidden Markov chain models we find an upper bound to the rate using results from Information Theory. We give sufficient conditions on the penalty term to avoid overestimation and underestimation of the order. Examples of penalty terms that generate strongly consistent estimators are also given.