On the finite sample breakdown points of redescending M-estimates of location

The finite sample breakdown points of scale equivariate redescending M-estimates of location are studied. In particular, a simple lower bound for the finite sample breakdown point of redescending M-estimates of location is given whenever the M-estimate of location is defined using the median absolute deviation about the median (MAD) as a scaling term. This lower bound is close to 0.49 for many common cases and depends on the configuration of the "good" data only through breakdown point of the MAD.