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June, 1995 On Estimating Mixing Densities in Discrete Exponential Family Models
Cun-Hui Zhang
Ann. Statist. 23(3): 929-945 (June, 1995). DOI: 10.1214/aos/1176324629

Abstract

This paper concerns estimating a mixing density function $g$ and its derivatives based on iid observations from $f(x) = \int f(x \mid \theta)g(\theta)d\theta$, where $f(x \mid \theta)$ is a known exponential family of density functions with respect to the counting measure on the set of nonnegative integers. Fourier methods are used to derive kernel estimators, upper bounds for their rate of convergence and lower bounds for the optimal rate of convergence. If $f(x \mid \theta_0) \geq \varepsilon^{x + 1} \forall x$, for some positive numbers $\theta_0$ and $\varepsilon$, then our estimators achieve the optimal rate of convergence $(\log n)^{-\alpha + m}$ for estimating the $m$th derivative of $g$ under a Lipschitz condition of order $\alpha > m$. The optimal rate of convergence is almost achieved when $(x!)^\beta f(x \mid \theta_0) \geq \varepsilon^{x + 1}$. Estimation of the mixing distribution function is also considered.

Citation

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Cun-Hui Zhang. "On Estimating Mixing Densities in Discrete Exponential Family Models." Ann. Statist. 23 (3) 929 - 945, June, 1995. https://doi.org/10.1214/aos/1176324629

Information

Published: June, 1995
First available in Project Euclid: 11 April 2007

zbMATH: 0841.62027
MathSciNet: MR1345207
Digital Object Identifier: 10.1214/aos/1176324629

Subjects:
Primary: 62G05
Secondary: 62G20

Rights: Copyright © 1995 Institute of Mathematical Statistics

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Vol.23 • No. 3 • June, 1995
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