Abstract Statistical Estimation and Modern Harmonic Analysis

Abstract

There is a considerable body of literature in the field of mathematical statistics to evaluate the minimax risk and to describe optimal and near-optimal estimators \hat{f} under various assumptions - the articles [2], [22], [30] are good starting points. This literature focuses on the question: What is the best way to recover f if all we know is that f has certain smoothness properties? In the author's view, modern harmonic analysis makes it possible to recast the problems of this literature in a more modular form, separating out key results into two components, one falling in the domain of harmonic analysis and one in the domain of statistical decision theory. This separation makes it possible for statisticians to avoid re-inventing the wheel; for solving the harmonic analysis part of their question they can fully exploit recent advances from modern harmonic analysis, rather than doing an elementary and possibly inadequate job from scratch. This separation also makes it possible for statisticians to do what they do best, and which no-one else will do for them: namely, solve problems of statistical decision theory. In the author's opinion, this separation will also suggest new questions in statistics; for example, if mathematical statisticians are not making use of all the tools of modern harmonic analysis, why not? Is this a sign that there are new problems they could be attacking and are not yet doing so? And so on.