A Consistent Estimator for the Identification of Finite Mixtures

Abstract

Henry Teicher [10] has initiated a systematic study called "identifiability of finite mixtures" (these terms to be defined in Section 1) which has significance in several areas of statistics. [10] gives a sufficiency condition that a family $\mathscr{F}$ of cdf's (cumulative distribution functions) generate identifiable finite mixtures, and consequently establishes that finite mixtures of the one-dimensional Gaussian or gamma families are identifiable. From [9] it is known that the Poisson family generates identifiable finite mixtures, and the binomial and uniform families do not. In [11], Teicher proves that the class of mixtures of $n$ products of any identifiable one-dimensional family is likewise identifiable (and that the analogous statement for finite mixtures is valid). Spragins and I have shown [13] that the finite mixtures on a family of cdf's is identifiable if and only if $\mathscr{F}$ is linearly independent in its span over the real numbers, and that $\mathscr{F}$ generates identifiable finite mixtures if $\mathscr{F}$ is any of the following: the $n$-dimensional normal family, the union of the $n$-dimensional normal family and the family of $n$ products of one dimensional exponential distributions, the Cauchy family, the negative binomial family, and the translation parameter family generated by any one dimensional cdf. (In this last case, our proof directly generalizes to any $n$-dimensional translation parameter family.) In view of the fact that many of the important distribution families have been seen to give identifiable finite mixtures, it would seem appropriate to seek methods for performing this identification, and therefore the intention of this paper is to reveal (Section 2) a general algorithm for construction of a consistent estimator. In Section 3 we demonstrate that the algorithm is effective for all the identifiable families mentioned above. Our results, in addition to having application to an interesting problem in communication theory [6], can be used to extend the empiric Bayes approach to a certain type of decision problem. Section 4 will discuss the details of this application.