Asymptotic Properties of the Maximum Likelihood Estimate of an Unknown Parameter of a Discrete Stochastic Process

Abstract

Asymptotic properties of maximum likelihood estimates have been studied so far mainly in the case of independent observations. In this paper the case of stochastically dependent observations is considered. It is shown that under certain restrictions on the joint probability distribution of the observations the maximum likelihood equation has at least one root which is a consistent estimate of the parameter $\theta$ to be estimated. Furthermore, any root of the maximum likelihood equation which is a consistent estimate of $\theta$ is shown to be asymptotically efficient. Since the maximum likelihood estimate is always a root of the maximum likelihood equation, consistency of the maximum likelihood estimate implies its asymptotic efficiency.