Consequences of Higher Order Asymptotics for the MSE of M-estimators on Neighborhoods

Abstract In Ruckdeschel (2010a), we derive an asymptotic expansion of the max-imal mean squared error (MSE) of location M-estimators on suitably thinned out, shrinking gross error neighborhoods. In this paper, we compile several consequences of this result: With the same techniques as 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. For the MSE problem, we tackle the problem of second order robust optimality: In the symmetric case, we find the second order op-timal 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. (2008) to our second order setup. Among all risk maximizing con-taminations we determine a “most innocent ” one. This way we quantify the “limits of detectability”in Huber (1997)’s definition for the purposes of robustness. Keywords higher order asymptotics · location M-estimator · second order optimality · minimax radius · cniper contaminatio