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

TL;DR

This paper explores higher order asymptotics of the mean squared error of M-estimators under contamination, revealing second order optimal scores and quantifying robustness limits.

Contribution

It derives higher order risk expressions and identifies second order optimal scores with smaller clipping heights, advancing robustness analysis of M-estimators.

Findings

Second order optimal scores are of Hampel form with smaller clipping heights.

Quantification of the limits of detectability in contamination scenarios.

Higher order risk expressions improve understanding of M-estimator robustness.

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.