Robust Statistics, 2nd Edition

4Linear Regression 1

4.1 Introduction

In this chapter we begin the discussion of the estimation of the parameters of linear regression models, which will be pursued in the next chapter. M‐estimators for regression are developed in the same way as for location. In this chapter we deal with fixed (nonrandom) predictors. Recall that our estimators of choice for location were redescending M‐estimators using the median as starting point and the MAD as dispersion. Redescending estimators will also be our choice for regression. When the predictors are fixed and fulfill certain conditions that are satisfied in particular for analysis of variance models, monotone M‐estimators – which are easy to compute – are robust, and can be used as starting points to compute a redescending estimator. When the predictors are random, or when they are fixed but in some sense “unbalanced”, monotone estimators cease to be reliable, and the starting points for redescending estimators must be computed otherwise. This problem is considered in the next chapter.

We start with an example that shows the weakness of the least‐squares estimator.

Example 4.1

In an experiment on the speed of learning of rats (Bond, 1979), times were recorded for a rat to go through a shuttlebox in successive attempts. If the time exceeded 5s, the rat received an electric shock for the duration of the next attempt. The data are the number of shocks received and the average time for all attempts between shocks. Tables and figures ...