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Finite mixtures of distributions with common central moments Rennie, Robert Richard

Abstract

Let ℱ = {F} be a family of n-variate cumulative distribution functions (c.d.f.'s). If F₁...,F(k) belong to ℱ and P₁,...,P(k) are positive numbers that sum to 1, then the convex combination M(x₁ ,. . . , x(n)) = [formula omitted](x₁,...,x(n)) is called a finite mixture generated by ℱ. The F₁,…,F(k) are called the components of the mixture and the P₁,…,P(k) are called their weights, respectively. The mixture M(x₁,...,x(n)) is said to be identifiable with respect to ℱ if no other convex combination of a finite number of c.d.f.'s from ℱ will generate M(x₁,...,x(n)). We establish the identiflability of mixtures consisting of at most k components when the components belong to a family of univariate c.d.f.'s that have the following properties: (a) no two c.d.f.'s have the same mean; (b) each c.d.f. has the same r(th) central moment for r = 1,...,2k-1; and (c) the first 2k-1 central moments are finite. If the mixture and the 2k-1 central moments are known, a solution for the weights and means of the components is given. If a random sample is taken from the mixture, then asymptotically normal estimates of the weights and means are given, providing the 2k-1 central moments are known. Matrix mixtures are introduced and are found to be of use in estimating the density functions and c.d.f.'s of the components. In the .case of the above family, the estimates of the density functions are shown to have an asymptotically normal distribution. Consistent and least squares estimates are obtained for the component c.d.f.'s. We show that for multivariate mixtures identifiability of any one of the marginal mixtures implies the identifiability of the multivariate mixture, but not conversely. Finally, the univariate results are generalized to the multivariate case, and an example of the use of matrix mixtures is given.

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