Characterizations of Statistical Distributions: A Supplement to Recent Surveys

Abstract

A survey of characterizations of statistical distributions may be approached from at least three angles: (1) Classification according to distributions by collecting and describing various characterization theorems pertaining to each individual specific distribution. This is the approach found in comprehensive books dealing partially or exclusively with statistical distribution theory such as the Kendall-Stuart [48] and the Johnson-Kotz [43] volumes. (2) Classification according to probabilistic and statistical methods and properties leading to characterizations. Among these properties are behaviour of linear statistics, a form of characteristic functions, independence, regression properties, sufficiency, etc. This approach is chosen in the recently published Russian monograph by Kagan, Linnik and Rao [46] (the English translation by Ramachandran, Wiley, 1973), devoted to characterization theorems in mathematical statistics. (3) Classification from a purely mathematical point of view according to mathematical techniques utilized in the proofs of characterization theorems such as functional equations, Faa di Bruno's formula, theory of entire analytic functions, methods of differential geometry and so on. This approach is taken by Lukacs [71] [75], Tan [128], Linnik [69] and others in occasional survey articles appearing in mathematical journals, in Proceedings of Symposia and to a certain extent in the monograph by Laha and Lukacs [67]. Each of these approaches is designed for a different class of readers. The first is more suitable for practitioners and applied statisticians who are mainly interested in adapting characterization theorems to the practical problems at hand, such as devising statistical and probabilistic models of real world phenomena or testing hypotheses on stochastic populations. The second would appeal mostly to theoretical and mathematical statisticians who are seeking a better insight into the structure and properties of the basic concepts of statistics via their interaction with distributions. The third approach is more appropriate for pure (and possibly applied) mathematicians dealing with the development and application of mathematical disciplines and techniques in relation to problems in probability theory and statistics. The mere fact that the subject of characterization theorems lends itself so naturally to these three distinct methods of investigation attests to the rich and abundant range of ideas contained in this field of study and hence the availability of a variety of challenging and useful problems for both theoretical and applied statisticians as well as for mathematicians with an inclination towards mathematical statistics.