Asymptotic minimax properties of M-estimators of scale

We ask whether or not the saddlepoint property holds, for robust M-estimation of scale, in gross-errors and Kolmogorov neighbourhoods of certain distributions. This is of interest since the saddlepoint property implies the minimax property -- that the supremum of the asymptotic variance of an M-estimator is minimized by the maximum likelihood estimator for that member of the distributional class with minimum Fisher information. Our findings are exclusively negative -- the saddlepoint property fails in all cases investigated.