Maximum asymptotic variances of trimmed means under asymmetric contamination (Q1072295)

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 \(F=(1-\epsilon)F_ 0+\epsilon G,\) where \(F_ 0\) is a known distribution symmetric about 0, \(\epsilon\) is a 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.