High Breakdown Point and High Efficiency

Abstract

While it is known that for robustness concepts based on the influence function (or on shrinking contamination neighbourhoods) high robustness and high efficiency can be combined by constrained problems (see in particular Hampel et al. (1986) and Chapter 7) there exists a recent discussion whether there is a conflict between efficiency and high breakdown point. Morgenthaler (1991), Stefanski (1991) and Coakley et al. (1994) showed that estimators with positive breakdown point have very low efficiencies compared with the least squares estimator. Therefore Davies (1993, 1994) proposed for desirable properties of estimators mainly robustness properties and no efficiency property. But as Rousseeuw (1994) argued this depends on the assumption of outliers in the independent variables (x-variables, experimental conditions) and the assumption of equal variances at all independent variables and in particular at leverage points. Hence, the conflict appears only in artificial situations. If the independent variables are random with possible outliers then a better model will be a multivariate model and in such model there will be no conflict between efficiency and positive breakdown (see IIe (1994)). For fixed independent variables as appear in planned experiments there is also no conflict as Morgenthaler (1994) noticed. But there is a conflict between high breakdown point designs and high efficient designs, which is shown in this chapter. At first basing on results of Section 4.3 in Section 9.1 trimmed weighted L_p estimators and corresponding designs are derived which maximize the breakdown point.