Stefan Van Aelst

Similarities Between Location Depth and Regression Depth

ℝ In this paper we first explore the analogies between location depth and regression depth. The location depth of [Tukey (1975)] is a multivariate generalization of rank, and leads to a multivariate median known as the Tukey median or the deepest location. Regression depth was introduced in [Rousseeuw and Hubert (1999b)], and yields the deepest regression which is a new robust regression estimator. Based on the recent literature on depth, we compare several theoretical and computational aspects of depth and of the deepest fits in location and regression.

A Robust Method for Multivariate Regression

We introduce a new method for multivariate regression based on robust estimation of the location and scatter matrix of the joint response and explanatory variables. The resulting method has good equivariance properties and the same breakdown value as the initial estimator for location and scatter. We also derive a general expression for the influence function at elliptical distributions. We compute asymptotic variances and compare them to finite-sample efficiencies obtained by simulation.

An Algorithm for Deepest Multiple Regression

Deepest regression (DR) is a method for linear regression introduced by Rousseeuw and Hubert (1999). The DR is defined as the fit with largest regression depth relative to the data. DR is a robust regression method. We construct an approximate algorithm for fast computation of DR in more than two dimensions. We also construct simultaneous confidence regions for the true unknown parameters, based on bootstrapped estimates.

The Deepest Fit

Recently, Rousseeuw & Hubert (1996) defined the depth of a regression fit relative to the data. This concept of regression depth immediately leads to a new robust regression estimator which we call the deepest fit. Quite simply, it is the fit with largest depth. Therefore, it can be seen as a generalization of the univariate median. We construct an algorithm to compute the deepest fit in simple regression, and illustrate it with examples. For any bivariate data set Z n the deepest fit has depth at least n/3, and a breakdown value of at least 1/3. Around the deepest fit we construct depth envelopes which generalize the quantiles around the univariate median.