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

Abstract

In Ruckdeschel[10], we derive an asymptotic expansion of the maximal 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[68] and Rieder[80b], respectively. For the MSE problem, we tackle the problem of second order robust optimality: 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 LO(1/n). For the case of unknown contamination radius we generalize the minimax inefficiency introduced in Rieder et al. [08] 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[97]'s definition for the purposes of robustness.