SPADES and mixture models

Details SPADES and mixture models SPADES and mixture models

2009-01-14

arxiv.org/abs/0901.2044v2

Annals of Statistics 2010, Vol. 38, No. 4, 2525-2558

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/09-AOS790 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

10.1214/09-AOS790

This paper studies sparse density estimation via $\ell_1$ 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 \citebb09. 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.

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