The Change-of-Variance Curve and Optimal Redescending M-Estimators

Robust Regression by Means of S-Estimators

“…Probably one could obtain higher values of e for the same g* by using pfunctions based on hyperbolic tangent estimators (Hampel, Rousseeuw and Ronchetti 1981). In fact, we wonder what is the maximal efficiency e, given a certain value of g* • Our definition of S-estimators (1.9) could easily be gene- factors A ), then also the following methods are S-estimators: least squares; least absolute deviations, least p-th power deviations (Gentleman 1965), the method of Jaeckel (1972), least median of squares (Rousseeuw 1982) and least trimmed squares (Rousseeuw 1983).…”

“…8) is Another possibility is to take a p corresponding to the function ~ proposed by Hampel, Rousseeuw and Ronchetti (1981). In general, 1jI(x) = p' (x) will always be zero for Ixl > c because of condition (R2); such 1jI-functions are usually called "redescending".…”

Redescending M-estimators

“…By analogy with the influence function, the change-of-variance function and its sensitivity are defined as (Hampel et al, 1981). For M -estimators that satisfy ψ ∈ C 1 (R), the change-of-variance function is (Hampel et al, 1986) CVF…”

Bounded-Influence Inference in Regression.

“…The first one, the influence function, was introduced by Hampel (1974) The sencond tool is the change-of-variance function; see Hampel, Rousseeuw, Ronchetti (1981), Ronchetti and Rousseeuw (1983). It can be viewed as the derivative of the asymptotic covariance matrix of the estimator, and describes its infinitesimal stability.…”