Maximum Asymptotic Variances of Trimmed Means Under Asymmetric Contamination

Abstract

We consider the following problem arising in robust estimation theory: Find the maximum asymptotic variance of a trimmed mean used to estimate an unknown location parameter when the error distribution is subject to asymmetric contamination. The model for the error distribution is $F = (1 - \varepsilon)F_0 + \varepsilon G$, where $F_0$ is a known distribution symmetric about $0, \varepsilon$ is fixed proportion of contamination, and $G$ is an unknown and possibly asymmetric distribution. We prove, under the assumption that $F_0$ has a symmetric unimodal density function $f_0$, that the maximal asymptotic variance is obtained when $G$ places mass 1 at either $+\infty$ or $-\infty$. The key idea of the proof is first to maximize the asymptotic variance subject to the side conditions $F(a) = \alpha$ and $F(b) = 1 - \alpha$ when $a$ and $b$ are given.