Bounded Influence Rank Regression

SUMMARY When εi = yi – x′i β, it is known that minimizing ΣΣ|εi – εj| yields an estimate of regression that attains a bounded influence of the residual with 95% efficiency for the normal distribution. We show that introducing weights ΣΣbij|εi – εj| achieves bounded total influence with positive breakdown. Mallows weights in particular are optimally efficient under a predefined bound on the gross error sensitivity. A generalization of Mallows weights allows additional local stability against high leverage points. Two numerical examples illustrate the behaviour of the estimate.