## The Annals of Statistics

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 62G05: Estimation

Secondary: 62G20: Asymptotic properties

**Keywords**

Mixture mixing distribution kernel estimator rate of convergence Fourier transformation

#### Citation

Zhang, Cun-Hui. On Estimating Mixing Densities in Discrete Exponential Family Models. Ann. Statist. 23 (1995), no. 3, 929--945. doi:10.1214/aos/1176324629. https://projecteuclid.org/euclid.aos/1176324629