## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 8, Number 1 (2014), 302-327.

### Aggregation of affine estimators

Dong Dai, Philippe Rigollet, Lucy Xia, and Tong Zhang

#### Abstract

We consider the problem of aggregating a general collection of affine estimators for fixed design regression. Relevant examples include some commonly used statistical estimators such as least squares, ridge and robust least squares estimators. Dalalyan and Salmon [DS12] have established that, for this problem, exponentially weighted (EW) model selection aggregation leads to sharp oracle inequalities in expectation, but similar bounds in deviation were not previously known. While results [DRZ12] indicate that the same aggregation scheme may not satisfy sharp oracle inequalities with high probability, we prove that a weaker notion of oracle inequality for EW that holds with high probability. Moreover, using a generalization of the newly introduced $Q$-aggregation scheme we also prove sharp oracle inequalities that hold with high probability. Finally, we apply our results to universal aggregation and show that our proposed estimator leads simultaneously to all the best known bounds for aggregation, including $\ell_{q}$-aggregation, $q\in(0,1)$, with high probability.

#### Article information

**Source**

Electron. J. Statist., Volume 8, Number 1 (2014), 302-327.

**Dates**

First available in Project Euclid: 10 April 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1397134174

**Digital Object Identifier**

doi:10.1214/14-EJS886

**Mathematical Reviews number (MathSciNet)**

MR3192554

**Zentralblatt MATH identifier**

1348.62132

**Subjects**

Primary: 62G08: Nonparametric regression

Secondary: 62C20: Minimax procedures 62G05: Estimation 62G20: Asymptotic properties

**Keywords**

Aggregation affine estimators Gaussian mean oracle inequalities Maurey’s argument

#### Citation

Dai, Dong; Rigollet, Philippe; Xia, Lucy; Zhang, Tong. Aggregation of affine estimators. Electron. J. Statist. 8 (2014), no. 1, 302--327. doi:10.1214/14-EJS886. https://projecteuclid.org/euclid.ejs/1397134174