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April, 1977 Estimation of a Mixing Distribution Function
J. R. Blum, V. Susarla
Ann. Probab. 5(2): 200-209 (April, 1977). DOI: 10.1214/aop/1176995845

Abstract

Let $a = \{f(\cdot, \theta): \theta \in J\}, J$ an interval, be a family of univariate probability densities (with respect to Lebesgue measure) on an interval I. First, a necessary and sufficient condition is proved for $\mathcal{a}$ to be identifiable whenever $\mathcal{a} \subset C_0(J)$, the class of continuous functions on $J$ vanishing at $\infty$. If $f_G$ is a $G$-mixture of the densities in $a$ with $G$ unknown, an estimator $G_n$ based on $f_G$ and $\mathscr{B} = \{f(x, \bullet): x \in I\}$ is provided such that $G_n \rightarrow_w G$ under certain conditions on $a$. If $X_1, \cdots, X_n$ are i.i.d. random variables from $f_G$, an estimator $\hat{G}_n$ is provided such that $G_n(X_1, \cdots, X_n, \cdot) \rightarrow_w G(\bullet)$ almost surely under certain conditions on $a$ and $G$. Furthermore, it is shown that $|f_{G_n}(x) - f_G(x)| \rightarrow 0$ a.s. and in $L_2$ with rates like $O(n^{-C}) (C > 0)$ under certain conditions on the density estimator $\hat{f}_G(x)$ involved in the definition of $\hat{G}_n$. The conditions of various theorems are verified in the case of location parameter and scale parameter families of densities.

Citation

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J. R. Blum. V. Susarla. "Estimation of a Mixing Distribution Function." Ann. Probab. 5 (2) 200 - 209, April, 1977. https://doi.org/10.1214/aop/1176995845

Information

Published: April, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0358.60040
MathSciNet: MR445703
Digital Object Identifier: 10.1214/aop/1176995845

Subjects:
Primary: 60F05
Secondary: 62099

Rights: Copyright © 1977 Institute of Mathematical Statistics

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Vol.5 • No. 2 • April, 1977
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