Advanced

Nonparametric estimation of mixing densities for discrete distributions

Roueff, Francois LU and Rydén, Tobias LU (2005) In Annals of Statistics 33(5). p.2066-2108
Abstract
By a mixture density is meant a density of the form pi(mu) (.) = f pi(theta) (.) x mu(d theta), where (pi(theta))(theta Theta is an element of) is a family of probability densities and mu is a probability measure on Theta. We consider the problem of identifying the unknown part of this model, the mixing distribution A, from a finite sample of independent observations from pi(mu). 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... (More)
By a mixture density is meant a density of the form pi(mu) (.) = f pi(theta) (.) x mu(d theta), where (pi(theta))(theta Theta is an element of) is a family of probability densities and mu is a probability measure on Theta. We consider the problem of identifying the unknown part of this model, the mixing distribution A, from a finite sample of independent observations from pi(mu). 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 oil 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 LIS 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. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
projection, mixtures of discrete distributions, minimax efficiency, Poisson mixtures, estimator, universal estimator
in
Annals of Statistics
volume
33
issue
5
pages
2066 - 2108
publisher
Inst Mathematical Statistics
external identifiers
  • wos:000234092100004
  • scopus:30344445702
ISSN
0090-5364
DOI
10.1214/009053605000000381
language
English
LU publication?
yes
id
c4fa6345-8c32-42c2-aed5-092fb7b70878 (old id 210824)
date added to LUP
2007-08-20 13:54:01
date last changed
2017-08-27 05:14:48
@article{c4fa6345-8c32-42c2-aed5-092fb7b70878,
  abstract     = {By a mixture density is meant a density of the form pi(mu) (.) = f pi(theta) (.) x mu(d theta), where (pi(theta))(theta Theta is an element of) is a family of probability densities and mu is a probability measure on Theta. We consider the problem of identifying the unknown part of this model, the mixing distribution A, from a finite sample of independent observations from pi(mu). 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 oil 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 LIS 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.},
  author       = {Roueff, Francois and Rydén, Tobias},
  issn         = {0090-5364},
  keyword      = {projection,mixtures of discrete distributions,minimax efficiency,Poisson mixtures,estimator,universal estimator},
  language     = {eng},
  number       = {5},
  pages        = {2066--2108},
  publisher    = {Inst Mathematical Statistics},
  series       = {Annals of Statistics},
  title        = {Nonparametric estimation of mixing densities for discrete distributions},
  url          = {http://dx.doi.org/10.1214/009053605000000381},
  volume       = {33},
  year         = {2005},
}