Introduction to Rousseeuw (1984) Least Median of Squares Regression

Abstract

Following seminal papers by Box (1953) and Tukey (1960), which demonstrated the need for robust statistical procedures, the theory of robust statistics blossomed in the 1960s and 1970s. An early milestone was Huber's (1964) paper which introduced univariate M-estimators and the minimax asymptotic variance criterion. Hampel (1974) proposed the influence function of an estimator as a way to describe the effect of a single outlier. In order to measure the effect of several outliers, he introduced the breakdown value (Hampel, 1971) in a general and asymptotic setting. Donoho and Huber (1983) advocated a finite-sample version of the breakdown value, in line with Hodges's (1967) study in the univariate framework. Heuristically, the breakdown point is the largest percentage of ill-fitting data that a method can cope with. For a formal definition, see equation (2.1) of the reprinted Rousseeuw (1984).