Matstat seminar onsdagen 20 juni 2001 i MH:227

Francois Roueff
Non-parametric estimation of discrete exponential family mixtures

Abstract:

Let X be a discrete random variable distributed according to the following model:
for all integer k, P{X=k} = E[p(Y,k)], where {p(t,.),t>=0} is a family of discrete probability distributions 
and Y is a non-negative variable distributed according to an unknown distribution.  
Moreover, we assume that {p(t,.),t>=0} has a particular form including the discrete Poisson family parameterized  
by their intensity. We consider the problem of identifying the unknown part of this model, 
i.e. the mixing distribution, from a finite sample of independent observations of X. More precisely, 
we are interested in a classical non-parametric setting: the mixing distribution has a density function and
we estimate this function within classical function spaces. Usual methods of density estimation such
as Kernel estimators or Wavelet estimators are available for this purpose and it may be 
(and, in some cases, 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 extend and 
simplify such results and allow to conclude in the asymptotic minimax efficiency of the latter estimators in 
the Poisson case. Because the rates which appear in this setting are rather slow (logarithmic order), such 
theoretical results are crucial for practical purposes and strongly advocate the use of such estimators 
rather than the "traditional" density estimators. 

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