A Note on Truncation and Sufficient Statistics

Abstract

Generalizing earlier observations by Fisher and Hotelling, Tukey [1] showed that if a family of distributions admits a set of sufficient statistics, then the family obtained by truncation to a fixed set, or by a fixed selection, also admits the same set of sufficient statistics (this wording is Tukey's; we give a precise mathematical statement later). Tukey's proof assumed the relevant family of probability measures to be dominated by a fixed measure function and made use of the factorization theorem concerning sufficient statistics in this case. In the present short note we shall first re-prove Tukey's result without assuming domination (and, hence, without appealing to the factorization theorem). Then we shall show that, under general conditions, if a sufficient statistic has one or more of the properties of completeness, bounded completeness, or minimality, before truncation, then it preserves such after truncation. The treatment is on the lines of the abstract discussion of sufficient statistics given by Halmos and Savage [2]. We shall assume familiarity with the results given in this latter paper. For definitions of completeness, bounded completeness, and minimality, and for a discussion of the significance of these concepts we refer to Lehmann and Scheffe [3].