3RD-ORDER OPTIMUM PROPERTY OF MAXIMUM LIKELIHOOD ESTIMATOR

Let θ(n) denote the maximum likelihood estimator of a vector parameter, based on an i.i.d. sample of size n. The class of estimators θ(n) + n-1 q(θ(n)), with q running through a class of sufficiently smooth functions, is essentially complete in the following sense: For any estimator T(n) there exists q such that the risk of θ(n) + n-1 q(θ(n)) exceeds the risk of T(n) by an amount of order o(n-1) at most, simultaneously for all loss functions which are bounded, symmetric, and neg-unimodal. If q* is chosen such that θ(n) + n-1 q*(θ(n)) is unbiased up to o(n -1 2), then this estimator minimizes the risk up to an amount of order o(n-1) in the class of all estimators which are unbiased up to o(n -1 2). The results are obtained under the assumption that T(n) admits a stochastic expansion, and that either the distributions have-roughly speaking-densities with respect to the lebesgue measure, or the loss functions are sufficiently smooth. © 1978.