## The Annals of Statistics

- Ann. Statist.
- Volume 21, Number 2 (1993), 965-990.

### Bias-Robust Estimates of Regression Based on Projections

Ricardo A. Maronna and Victor J. Yohai

#### Abstract

A new class of bias-robust estimates of multiple regression is introduced. If $y$ and $x$ are two real random variables, let $T(y, x)$ be a univariate robust estimate of regression of $y$ on $x$ through the origin. The regression estimate $\mathbf{T}(y, \mathbf{x})$ of a random variable $y$ on a random vector $\mathbf{x} = (x_1,\cdots, x_p)'$ is defined as the vector $\mathbf{t} \in \mathfrak{R}^p$ which minimizes $\sup_{\|\mathbf{\lambda}\| = 1} \mid T(y - \mathbf{t'x, \lambda' x}) \mid s(\mathbf{\lambda'x})$, where $s$ is a robust estimate of scale. These estimates, which are called projection estimates, are regression, affine and scale equivariant. When the univariate regression estimate is $T(y, x) =$ median $(y/x)$, the resulting projection estimate is highly bias-robust. In fact, we find an upper bound for its maximum bias in a contamination neighborhood, which is approximately twice the minimum possible value of this maximum bias for any regression and affine equivariant estimate. The maximum bias of this estimate in a contamination neighborhood compares favorably with those of Rousseeuw's least median squares estimate and of the most bias-robust GM-estimate. A modification of this projection estimate, whose maximum bias for a multivariate normal with mass-point contamination is very close to the minimax bound, is also given. Projection estimates are shown to have a rate of consistency of $n^{1/2}$. A computational version of these estimates, based on subsampling, is given. A simulation study shows that its small sample properties compare very favorably to those of other robust regression estimates.

#### Article information

**Source**

Ann. Statist., Volume 21, Number 2 (1993), 965-990.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176349160

**Digital Object Identifier**

doi:10.1214/aos/1176349160

**Mathematical Reviews number (MathSciNet)**

MR1232528

**Zentralblatt MATH identifier**

0787.62037

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F35: Robustness and adaptive procedures

Secondary: 62J05: Linear regression

**Keywords**

Minimax bias regression robust estimates

#### Citation

Maronna, Ricardo A.; Yohai, Victor J. Bias-Robust Estimates of Regression Based on Projections. Ann. Statist. 21 (1993), no. 2, 965--990. doi:10.1214/aos/1176349160. https://projecteuclid.org/euclid.aos/1176349160