IDENTIFIABILITY OF MIXTURES OF POWER-SERIES DISTRIBUTIONS AND RELATED CHARACTERIZATIONS

THEOFANIS SAPATINAS

Department of Mathematical Statistics and Operational Research, Exeter University, Exeter EX4-4QE, U.K.

(Received January 28, 1994; revised October 24, 1994)

Abstract. The concept of the identifiability of mixtures of distributions is discussed and a sufficient condition for the identifiability of the mixture of a large class of discrete distributions, namely that of the power-series distributions, is given. Specifically, by using probabilistic arguments, an elementary and shorter proof of the Lüxmann-Ellinghaus’s (1987, Statist. Probab. Lett., 5, 375–378) result is obtained. Moreover, it is shown that this result is a special case of a stronger result connected with the Stieltjes moment problem. Some recent observations due to Singh and Vasudeva (1984, J. Indian Statist. Assoc., 22, 93–96) and Johnson and Kotz (1989, Ann. Inst. Statist. Math., 41, 13–17) concerning characterizations based on conditional distributions are also revealed as special cases of this latter result. Exploiting the notion of the identifiability of power-series mixtures, characterizations based on regression functions (posterior expectations) are obtained. Finally, multivariate generalizations of the preceding results have also been addressed.

Key words and phrases: Univariate and multivariate power-series distributions, mixtures of distributions, the moment problem, infinite divisibility, posterior expectations.