Role of Moments in Statistics and Probability

Abstract

The moments are the simplest characteristics of? any distribution and it is remarkable that they have been used efficiently in inference problems for many decades, since the time of Karl Pearson up to now. All this was possible only because there is a rich and attractive theory of probability/statistical distributions. An important part of this theory is based on the moments.? Erroneously, some 10 years ago, Professor Terence Speed (Department of Statistics, Berkeley) published in the Bulletin of IMS an article titled: “The Moments are Death”.? No comments.? Among the main topics to be discussed in the talk are: (a) clear general picture of available and checkable (!) conditions about uniqueness and non-uniqueness of a distribution by its moments; (b) two anniversaries: 100 years of Hardy’s condition and 20 years of condition (L); (c) usefulness of Frechet-Shohat limit theorem for both, theory and applications; (d) recent developments: using the rate of growth of the moments, dealing with discrete distributions, treating non-linear Box-Cox transformations of random data, recovering an unknown distribution or? density from its moments, constructing Stochastic Differential Equations with prescribed marginal properties, etc. (e) If time allows, some open questions will be outlined.?