Convolution Theorem and Asymptotic Efficiency

Abstract

The optimality of the maximum likelihood estimate developed in Chapters 8 and 9 is in terms of the asymptotic variance of the solution of estimating equations. However, this optimality goes much deeper. In this chapter the maximum likelihood estimate, as well as estimates that are asymptotically equivalent to it, are shown to have the smallest asymptotic variance among all regular estimates, which is a much wider class of estimates than solutions of estimating equations. The theory underlying this general result – the framework of Local Asymptotic Normality and the Convolution Theorem – is systematically developed. This is an amazingly logical system that leads to far-reaching results with a small set of assumptions. Some techniques introduced in this chapter, such as Le Cam’s third lemma and the convolution theorem, will also be useful for developing local alternative distributions for asymptotic hypothesis tests in Chapter 11.