Partially Complete Sufficient Statistics Are Jointly Complete

Abstract

The theorem formulated in the title (surprisingly, the result has been overlooked for many years) complements the theory of sufficient statistics in the part related to Lehmann--Scheffé completeness. Two proofs of the main result are given: The first, analytical, follows Landers and Rogge [Scand. J. Statist., 3 (1976), p. 139], while the second, purely statistical, uses the theory of optimal unbiased estimation in a relatively little known version completed by Schmetterer and Strasser [Anz. Österreich. Akad. Wiss. Math.-Naturwiss. Kl., 1974, no. 6, pp. 59--66]. Connections to previous results are discussed and illustrative examples are presented.