Higher order asymptotics for the MSE of M-estimators on shrinking neighborhoods

In the setup of shrinking neighborhoods about an ideal central model, Rieder (1994) determines the as. linear estimator minimaxing MSE on these neighborhoods. We address the question to which degree this as. optimality carries over to finite sample size. We consider estimation of a one-dim. location parameter by means of M-estimators Sn with monotone influence curve ψ . Using Donoho and Huber (1983)’s finite sample breakdown point ε0 for Sn , we define thinned out convex contamination balls Q̃n(r; ε0) of radius r/ √ n about the ideal distribution.This modification is negligible exponentially, but suffices to establish uniform higher order asymptotics for the MSE of the kind max Qn∈Q̃n(r;ε0) nMSE(Sn, Qn) = r 2 supψ + Eidψ 2 + r √ n A1 + 1 n A2 + o( 1 n ), where A1 , A2 are constants depending on ψ and r . Moreover, we essentially characterize contaminations generating maximal MSE up to o(n−1) . Our results are confirmed empirically by simulations as well as numerical evaluations of the risk. With the techniques used for the MSE , we determine higher order expressions for the risk based on over-/undershooting probabilities as in Huber (1968) and Rieder (1980), respectively. In the symmetric case, we find the second order optimal scores again of Hampel form, but to an O(n−1/2) -smaller clipping height c than in first order asymptotics. This smaller c improves MSE only by O(n−1) . For the case of unknown contamination radius we generalize the minimax inefficiency introduced in Rieder et al. (2001) to our second order setup. Among all risk maximizing contaminations we determine a “most innocent” one. This way we quantify the “limits of detectability”in Huber (1997)’s definition for the purposes of robustness. ∗..