The Annals of Statistics

SPADES and mixture models

Florentina Bunea, Alexandre B. Tsybakov, Marten H. Wegkamp, and Adrian Barbu

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Abstract

This paper studies sparse density estimation via 1 penalization (SPADES). We focus on estimation in high-dimensional mixture models and nonparametric adaptive density estimation. We show, respectively, that SPADES can recover, with high probability, the unknown components of a mixture of probability densities and that it yields minimax adaptive density estimates. These results are based on a general sparsity oracle inequality that the SPADES estimates satisfy. We offer a data driven method for the choice of the tuning parameter used in the construction of SPADES. The method uses the generalized bisection method first introduced in [10]. The suggested procedure bypasses the need for a grid search and offers substantial computational savings. We complement our theoretical results with a simulation study that employs this method for approximations of one and two-dimensional densities with mixtures. The numerical results strongly support our theoretical findings.

Article information

Source
Ann. Statist., Volume 38, Number 4 (2010), 2525-2558.

Dates
First available in Project Euclid: 11 July 2010

Permanent link to this document
https://projecteuclid.org/euclid.aos/1278861256

Digital Object Identifier
doi:10.1214/09-AOS790

Mathematical Reviews number (MathSciNet)
MR2676897

Zentralblatt MATH identifier
1198.62025

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 62C20: Minimax procedures 62G05: Estimation 62G20: Asymptotic properties

Keywords
Adaptive estimation aggregation lasso minimax risk mixture models consistent model selection nonparametric density estimation oracle inequalities penalized least squares sparsity statistical learning

Citation

Bunea, Florentina; Tsybakov, Alexandre B.; Wegkamp, Marten H.; Barbu, Adrian. SPADES and mixture models. Ann. Statist. 38 (2010), no. 4, 2525--2558. doi:10.1214/09-AOS790. https://projecteuclid.org/euclid.aos/1278861256


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