Non-parametric estimation of mixing densities for discrete distributions
2005 (English)In: Annals of Statistics, ISSN 0090-5364, Vol. 33, no 5, 2066-2108 p.Article in journal (Refereed) Published
By a mixture density is meant a density of the form πμ(⋅)=∫πθ(⋅)×μ(dθ), where (πθ)θ∈Θ is a family of probability densities and μ is a probability measure on Θ. We consider the problem of identifying the unknown part of this model, the mixing distribution μ, from a finite sample of independent observations from πμ. Assuming that the mixing distribution has a density function, we wish to estimate this density within appropriate function classes. A general approach is proposed and its scope of application is investigated in the case of discrete distributions. Mixtures of power series distributions are more specifically studied. Standard methods for density estimation, such as kernel estimators, are available in this context, and it has been shown that these methods are rate optimal or almost rate optimal in balls of various smoothness spaces. For instance, these results apply to mixtures of the Poisson distribution parameterized by its mean. Estimators based on orthogonal polynomial sequences have also been proposed and shown to achieve similar rates. The general approach of this paper extends and simplifies such results. For instance, it allows us to prove asymptotic minimax efficiency over certain smoothness classes of the above-mentioned polynomial estimator in the Poisson case. We also study discrete location mixtures, or discrete deconvolution, and mixtures of discrete uniform distributions.
Place, publisher, year, edition, pages
2005. Vol. 33, no 5, 2066-2108 p.
Mixtures of discrete distributions, minimax efficiency, projection estimator, universal estimator, Poisson mixtures
Probability Theory and Statistics
IdentifiersURN: urn:nbn:se:kth:diva-61530DOI: 10.1214/009053605000000381ISI: 000234092100004OAI: oai:DiVA.org:kth-61530DiVA: diva2:479321
QC 201201262012-01-172012-01-172012-01-26Bibliographically approved