Identifiability of mixtures of power-series distributions and related characterizations

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.

[1]  J. Shohat,et al.  The problem of moments , 1943 .

[2]  Herbert Robbins,et al.  Mixture of Distributions , 1948 .

[3]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[4]  N. L. Johnson,et al.  Uniqueness of a result in the theory of accident proneness , 1957 .

[5]  James E. Mosimann,et al.  On the compound negative multinomial distribution and correlations among inversely sampled pollen counts , 1963 .

[6]  O. Barndorff-Nielsen.  Identifiability of mixtures of exponential families , 1965 .

[7]  G. Patil,et al.  Identifiability of countable mixtures of discrete probability distributions using methods of infinite matrices , 1966, Mathematical Proceedings of the Cambridge Philosophical Society.

[8]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[9]  N. L. Johnson,et al.  Note on a Uniqueness Relation in Certain Accident Proneness Models , 1967 .

[10]  S. Yakowitz,et al.  On the Identifiability of Finite Mixtures , 1968 .

[11]  N. Akhiezer,et al.  The Classical Moment Problem. , 1968 .

[12]  W. A. Ericson.  A Note on the Posterior Mean of a Population Mean , 1969 .

[13]  M. Sankaran,et al.  275. Note: The Discrete Poisson-Lindley Distribution , 1970 .

[14]  P. Diaconis,et al.  Conjugate Priors for Exponential Families , 1979 .

[15]  Masaaki Sibuya,et al.  Generalized hypergeometric, digamma and trigamma distributions , 1979 .

[16]  Martin D. Fraser,et al.  Identifiability of Finite Mixtures of Von Mises Distributions , 1981 .

[17]  G. M. Tallis,et al.  Identifiability of mixtures , 1982, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[18]  John T. Kent,et al.  Identifiability of Finite Mixtures for Directional Data , 1983 .

[19]  U. Lüxmann-Ellinghaus,et al.  On the identifiability of mixtures of infinitely divisible power series distributions , 1987 .

[20]  T. Cacoullos.  Characterizing priors by posterior expectations in multiparameter exponential families , 1987 .

[21]  Characterization based on conditional distributions , 1989 .

[22]  On characterizations of exponential distributions , 1989 .

[23]  B. Rao.  Identifiability for Mixtures of Distributions , 1992 .

[24]  T. Rao,et al.  Identifiability in stochastic models : characterization of probability distributions , 1994 .