Higher Order Expansion for the MSE of M-estimators on shrinking neighborhoods

TL;DR

This paper derives a higher order expansion for the maximum mean squared error of M-estimators under shrinking contamination neighborhoods, providing insights into their robustness and asymptotic behavior.

Contribution

It introduces a higher order asymptotic expansion for the MSE of M-estimators on shrinking neighborhoods, including constants and conditions for robustness.

Findings

The expansion includes terms up to 1/n with explicit constants.

M-estimators remain square integrable under contamination neighborhoods.

Results are validated through simulations and numerical risk evaluations.

Abstract

We consider estimation of a one-dimensional location parameter by means of M-estimators S_n with monotone influence curve psi. For growing sample size n, on suitably thinned out convex contamination ball BQ_n of shrinking radius r/sqrt(n) about the ideal distribution, we obtain an expansion of the asymptotic maximal mean squared error MSE of form r^2 sup psi^2 + E_{id} psi^2 + r/sqrt(n) A_1 + 1/n A_2 + o(1/n), where A_1, A_2 are constants depending on psi and r. Hence S_n not only is uniformly (square) integrable in n (in the ideal model) but also on BQ_n, which is not self-evident. For this result, the thinning of the neighborhoods, by a breakdown-driven, sample-wise restriction, is crucial, but exponentially negligible. Moreover, our results essentially characterize contaminations generating maximal MSE up to o(1/n). Our results are confirmed empirically by simulations as well as numerical evaluations of the risk.