## The Annals of Statistics

- Ann. Statist.
- Volume 35, Number 6 (2007), 2421-2449.

### Estimation and confidence sets for sparse normal mixtures

T. Tony Cai, Jiashun Jin, and Mark G. Low

#### Abstract

For high dimensional statistical models, researchers have begun to focus on situations which can be described as having relatively few *moderately large* coefficients. Such situations lead to some very subtle statistical problems. In particular, Ingster and Donoho and Jin have considered a sparse normal means testing problem, in which they described the precise demarcation or *detection boundary*. Meinshausen and Rice have shown that it is even possible to estimate consistently the fraction of nonzero coordinates on a subset of the detectable region, but leave unanswered the question of exactly in which parts of the detectable region consistent estimation is possible.

In the present paper we develop a new approach for estimating the fraction of nonzero means for problems where the nonzero means are moderately large. We show that the detection region described by Ingster and Donoho and Jin turns out to be the region where it is possible to consistently estimate the expected fraction of nonzero coordinates. This theory is developed further and minimax rates of convergence are derived. A procedure is constructed which attains the optimal rate of convergence in this setting. Furthermore, the procedure also provides an honest lower bound for confidence intervals while minimizing the expected length of such an interval. Simulations are used to enable comparison with the work of Meinshausen and Rice, where a procedure is given but where rates of convergence have not been discussed. Extensions to more general Gaussian mixture models are also given.

#### Article information

**Source**

Ann. Statist., Volume 35, Number 6 (2007), 2421-2449.

**Dates**

First available in Project Euclid: 22 January 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1201012967

**Digital Object Identifier**

doi:10.1214/009053607000000334

**Mathematical Reviews number (MathSciNet)**

MR2382653

**Zentralblatt MATH identifier**

1360.62113

**Subjects**

Primary: 62G05: Estimation

Secondary: 62G20: Asymptotic properties 62G32: Statistics of extreme values; tail inference

**Keywords**

Confidence lower bound estimating fraction higher criticism minimax estimation optimally adaptive sparse normal mixture

#### Citation

Cai, T. Tony; Jin, Jiashun; Low, Mark G. Estimation and confidence sets for sparse normal mixtures. Ann. Statist. 35 (2007), no. 6, 2421--2449. doi:10.1214/009053607000000334. https://projecteuclid.org/euclid.aos/1201012967