## The Annals of Statistics

### On Estimating Mixing Densities in Discrete Exponential Family Models

Cun-Hui Zhang

#### 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.

#### Article information

Source
Ann. Statist., Volume 23, Number 3 (1995), 929-945.

Dates
First available in Project Euclid: 11 April 2007

https://projecteuclid.org/euclid.aos/1176324629

Digital Object Identifier
doi:10.1214/aos/1176324629

Mathematical Reviews number (MathSciNet)
MR1345207

Zentralblatt MATH identifier
0841.62027

JSTOR