Higher order asymptotics for the MSE of the sample median on shrinking neighborhoods

TL;DR

This paper derives higher order asymptotic expansions for the maximal MSE of the sample median under shrinking neighborhoods, refining first order results by directly expanding the risk without approximating distributions.

Contribution

It introduces a novel higher order asymptotic expansion of the sample median's MSE on shrinking neighborhoods, directly expanding the risk rather than distribution functions.

Findings

Asymptotic expansion in powers of n^{-1/2} for the MSE

Comparison with simulation and exact MSE calculations

Refinement of first order asymptotics for contaminated data

Abstract

We provide an asymptotic expansion of the maximal mean squared error (MSE) of the sample median to be attained on shrinking gross error neighborhoods about an ideal central distribution. More specifically, this expansion comes in powers of n^{-1/2}, for n the sample size, and uses a shrinking rate of n^{-1/2} as well. This refines corresponding results of first order asymptotics to be found in Rieder[94]. In contrast to usual higher order asymptotics, we do not approximate distribution functions (or densities) in the first place, but rather expand the risk directly. Our results are illustrated by comparing them to the results of a simulation study and to numerically evaluated exact MSE's in both ideal and contaminated situation.