Nonparametric 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,
2002
ISSN 14039338

Abstract:

Let $\{\pi_\theta(\cdot)\}$ be a family of probability distributions on the
nonnegative 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 nonparametric 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
