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March, 1960 On the Mixture of Distributions
Henry Teicher
Ann. Math. Statist. 31(1): 55-73 (March, 1960). DOI: 10.1214/aoms/1177705987


If $\mathcal{F} = \{F\}$ is a family of distribution functions and $\mu$ is a measure on a Borel Field of subsets of $\mathcal{F}$ with $\mu(\mathcal{F}) = 1$, then $\int F(\cdot) d\mu (F)$ is again a distribution function which is called a $\mu$-mixture of $\mathcal{F}$. In Section 2, convergence questions when either $F_n$ or $\mu_k$ (or both) tend to limits are dealt with in the case where $\mathcal{F}$ is indexed by a finite number of parameters. In Part 3, mixtures of additively closed families are considered and the class of such $\mu$-mixtures is shown to be closed under convolution (Theorem 3). In Section 4, a sufficient as well as necessary conditions are given for a $\mu$-mixture of normal distributions to be normal. In the case of a product-measure mixture, a necessary and sufficient condition is obtained (Theorem 7). Generation of mixtures is discussed in Part 5 and the concluding remarks of Section 6 link the problem of mixtures of Poisson distributions to a moment problem.


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Henry Teicher. "On the Mixture of Distributions." Ann. Math. Statist. 31 (1) 55 - 73, March, 1960.


Published: March, 1960
First available in Project Euclid: 27 April 2007

zbMATH: 0107.13501
MathSciNet: MR121825
Digital Object Identifier: 10.1214/aoms/1177705987

Rights: Copyright © 1960 Institute of Mathematical Statistics

Vol.31 • No. 1 • March, 1960
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