Nonparametric estimation of mixing densities for discrete distributions
(2005) In Annals of Statistics 33(5). p.20662108 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 abovementioned 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:
https://lup.lub.lu.se/record/210824
 author
 Roueff, Francois ^{LU} and Rydén, Tobias ^{LU}
 organization
 publishing date
 2005
 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
 Institute of Mathematical Statistics
 external identifiers

 wos:000234092100004
 scopus:30344445702
 ISSN
 00905364
 DOI
 10.1214/009053605000000381
 language
 English
 LU publication?
 yes
 id
 c4fa63458c3242c2aed5092fb7b70878 (old id 210824)
 date added to LUP
 20160401 15:52:39
 date last changed
 20210420 02:13:30
@article{c4fa63458c3242c2aed5092fb7b70878, 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 abovementioned 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 = {00905364}, language = {eng}, number = {5}, pages = {20662108}, publisher = {Institute of Mathematical Statistics}, series = {Annals of Statistics}, title = {Nonparametric estimation of mixing densities for discrete distributions}, url = {http://dx.doi.org/10.1214/009053605000000381}, doi = {10.1214/009053605000000381}, volume = {33}, year = {2005}, }