SOME THEOREMS AND SUFFICIENCY CONDITIONS FOR THE MAXIMUM-LIKELIHOOD ESTIMATOR OF AN UNKNOWN PARAMETER IN A SIMPLE MARKOV CHAIN

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Abstract
The paper begins with proofs of the usual theorems for the optimum properties of the maximum-likelihood estimator of an unknown parameter θ which defines the transition probabilities pij(θ) of a simple ergodic Markov chain. By an ergodic chain is meant one for which, not only is the final chain stationary, but also all possible initial states remain permanently available; these conditions are sufficient to prove that the maximum-likelihood estimator is consistent, and asymptotically normally distributed. The paper proceeds to establish the form of the transition probabilities pij(θ) which admit a sufficient estimator of θ. To do this, the form of the likelihood function admitting a sufficient estimator when the parent distribution is discrete is first derived: this is used to obtain the form of the probabilities Pij(θ) for a multinomial distribution admitting a sufficient estimator of θ. and the result is finally generalized for the transition probabilities pij(θ) of the simple ergodic Markov chain. The paper closes with an examination of possible forms for the matrix p of transition probabilities pij(θ), and these are illustrated with simple examples for Markov chains with two and three states.

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