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IDENTIFIABILITY OF MIXTURES OF POWER-SERIES

DISTRIBUTIONS AND RELATED CHARACTERIZATIONS

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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.

**Source**
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