Identifiability of mixtures of power-series distributions and related characterizations

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. © 1995 The Institute of Statistical Mathematics.