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Biometrika 1955 42(3-4):342-359; doi:10.1093/biomet/42.3-4.342
© 1955 by Biometrika Trust
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SOME THEOREMS AND SUFFICIENCY CONDITIONS FOR THE MAXIMUM-LIKELIHOOD ESTIMATOR OF AN UNKNOWN PARAMETER IN A SIMPLE MARKOV CHAIN

J. GANI

Australian National University Canberra, A.C.T.

The paper begins with proofs of the usual theorems for the optimum properties of the maximum-likelihood estimator of an unknown parameter {theta} which defines the transition probabilities pij({theta}) 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({theta}) which admit a sufficient estimator of {theta}. 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({theta}) for a multinomial distribution admitting a sufficient estimator of {theta}. and the result is finally generalized for the transition probabilities pij({theta}) of the simple ergodic Markov chain.

The paper closes with an examination of possible forms for the matrix p of transition probabilities pij({theta}), and these are illustrated with simple examples for Markov chains with two and three states.


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