Breakdown points and variation exponents of robust $M$-estimators in linear models

The breakdown point behavior of $M$-estimators in linear models with fixed designs, arising from planned experiments or qualitative factors, is characterized. Particularly, this behavior at fixed designs is quite different from that at designs which can be corrupted by outliers, the situation prevailing in the literature. For fixed designs, the breakdown points of robust $M$-estimators (those with bounded derivative of the score function), depend on the design and the variation exponent (index) of the score function. This general result implies that the highest breakdown point within all regression equivariant estimators can be attained also by certain $M$-estimators: those with slowly varying score function, like the Cauchy or slash maximum likelihood estimator. The $M$-estimators with variation exponent greater than 0, like the $L_1$ or Huber estimator, exhibit a consider-ably worse breakdown point behavior.