The Annals of Statistics

Approximation of conditional densities by smooth mixtures of regressions

Andriy Norets

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Abstract

This paper shows that large nonparametric classes of conditional multivariate densities can be approximated in the Kullback–Leibler distance by different specifications of finite mixtures of normal regressions in which normal means and variances and mixing probabilities can depend on variables in the conditioning set (covariates). These models are a special case of models known as “mixtures of experts” in statistics and computer science literature. Flexible specifications include models in which only mixing probabilities, modeled by multinomial logit, depend on the covariates and, in the univariate case, models in which only means of the mixed normals depend flexibly on the covariates. Modeling the variance of the mixed normals by flexible functions of the covariates can weaken restrictions on the class of the approximable densities. Obtained results can be generalized to mixtures of general location scale densities. Rates of convergence and easy to interpret bounds are also obtained for different model specifications. These approximation results can be useful for proving consistency of Bayesian and maximum likelihood density estimators based on these models. The results also have interesting implications for applied researchers.

Article information

Source
Ann. Statist. Volume 38, Number 3 (2010), 1733-1766.

Dates
First available in Project Euclid: 24 March 2010

Permanent link to this document
http://projecteuclid.org/euclid.aos/1269452653

Digital Object Identifier
doi:10.1214/09-AOS765

Mathematical Reviews number (MathSciNet)
MR2662358

Zentralblatt MATH identifier
05712437

Subjects
Primary: 62G07: Density estimation
Secondary: 41A30: Approximation by other special function classes

Keywords
Finite mixtures of normal distributions smoothly mixing regressions mixtures of experts Bayesian conditional density estimation

Citation

Norets, Andriy. Approximation of conditional densities by smooth mixtures of regressions. Ann. Statist. 38 (2010), no. 3, 1733--1766. doi:10.1214/09-AOS765. http://projecteuclid.org/euclid.aos/1269452653.


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