Non-parametric Estimation of Mixing Densities in a Class of Discrete Distributions

Francois Roueff and Tobias Rydén

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,

ISSN 1403-9338
Let $\{\pi_\theta(\cdot)\}$ be a family of probability distributions on the non-negative integers, indexed by a real parameter $\theta\geq0$. By a mixture of this family is meant a distribution of the form $\pi_\mu(k)=\int\pi_\theta(k)\,d\mu(\theta)$, where $\mu$ is a probability measure on $[0,\infty)$. We assume that $\pi_\theta$ has a particular form including the Poisson family parameterized by its mean and the negative binomial distribution.  We consider the problem of identifying the unknown part of this model, i.e. the mixing distribution $\mu$, from a finite sample of independent observations from $\pi_\mu$. More precisely, we are interested in a classical non-parametric setting and assume that the mixing distribution has a density function and we estimate this function within appropriate function spaces. Usual methods of density estimation such as kernel estimators and wavelet estimators are available for this purpose and it has been shown that these methods are rate optimal in balls of various smoothness spaces. Other estimators based on orthogonal polynomial sequences have also been proposed and shown to achieve similar rates. Here we propose a simple formalism which extends and simplifies such results and allows us to prove asymptotic minimax efficiency of the latter estimators in the Poisson case. Because the rates which appear in this setting are rather slow, of logarithmic order, such theoretical results are crucial for practical purposes and strongly advocate estimators using orthogonal polynomials.
Key words:
Mixtures of discrete distributions, minimax efficiency, projection estimator, universal estimator, Poisson mixtures