On an optimal asymptotic property of the maximum likelihood estimator of a parameter from a stochastic process

This paper is concerned with the estimation of a parameter of a stochastic process on the basis of a single realization. It is shown, under suitable regularity conditions, that the maximum likelihood estimator is the best consistent asymptotically normal estimator in the sense of having minimum asymptotic variance. It also produces the best limiting probability of concentration in symmetric intervals. An application is given for the problem of estimating the mean of the offspring distribution in a Galton-Watson branching process.