On the explosion rate of maximum-bias functions

The maxbias function B_T() contains much information about the robustness properties of the estimate T. This function satisfies B_T(0)=0 and B_T()breakdown point of T. Hampel (1974)pioneered the study of the limiting behaviour of B_T(?) where ? → 0. He computed and optimized the rate γ at which B_T(?) approaches 0 when ? → 0. This rate is now called the contamination sensitivity of T, and constitutes one of the cornerstones of the theory of robustness. We show that much can also be learned from the study of the limiting behaviour of B_T(?) when ? → ?*. A new robustness measure, called the relative explosion rate, can be obtained by studying the limiting relative maxbias behaviour of two extimates when approaches their common breakdown point ?*. Like the contamination sensitivity, the relative explosion rate can be readily derived from the estimate's score function. General formulae are given for M-estimates of scale and S-, MM- and τ-estimates of regression. We also show that the maxbias behaviour for large ? is largely determined by the curvature of the estimate's score function near zero. This motivates our definition and study of the local order of a score function.