## The Annals of Statistics

- Ann. Statist.
- Volume 45, Number 5 (2017), 1920-1950.

### Detection and feature selection in sparse mixture models

Nicolas Verzelen and Ery Arias-Castro

#### Abstract

We consider Gaussian mixture models in high dimensions, focusing on the twin tasks of detection and feature selection. Under sparsity assumptions on the difference in means, we derive minimax rates for the problems of testing and of variable selection. We find these rates to depend crucially on the knowledge of the covariance matrices and on whether the mixture is symmetric or not. We establish the performance of various procedures, including the top sparse eigenvalue of the sample covariance matrix (popular in the context of Sparse PCA), as well as new tests inspired by the normality tests of Malkovich and Afifi [*J. Amer. Statist. Assoc.* **68** (1973) 176–179].

#### Article information

**Source**

Ann. Statist., Volume 45, Number 5 (2017), 1920-1950.

**Dates**

Received: May 2014

Revised: December 2015

First available in Project Euclid: 31 October 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1509436823

**Digital Object Identifier**

doi:10.1214/16-AOS1513

**Mathematical Reviews number (MathSciNet)**

MR3718157

**Zentralblatt MATH identifier**

06821114

**Subjects**

Primary: 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20] 62H15: Hypothesis testing

**Keywords**

Gaussian mixture models detection of mixtures feature selection for mixtures sparse mixture models the sparse eigenvalue problem projection tests based on moments

#### Citation

Verzelen, Nicolas; Arias-Castro, Ery. Detection and feature selection in sparse mixture models. Ann. Statist. 45 (2017), no. 5, 1920--1950. doi:10.1214/16-AOS1513. https://projecteuclid.org/euclid.aos/1509436823

#### Supplemental materials

- Supplement to “Detection and feature selection in sparse mixture models”. This supplement contains the proofs of the results.Digital Object Identifier: doi:10.1214/16-AOS1513SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.