Partially Complete Sufficient Statistics Are Jointly Complete
Abram Meerovich Kagan, Abram Kagan, Y Malinovsky, Y Malinovsky, Lutz Mattner, Lutz Mattner · Theory of Probability and Its Applications · 2014
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.