Minimax Variance M-Estimators in $\varepsilon$-Contamination Models

Abstract

In the framework of Huber's theory of robust estimation of a location parameter, minimax variance M-estimators are studied for error distributions with densities of the form $f(x) = (1 - \varepsilon)h(x) + \varepsilon g(x)$, where $g$ is unknown. A well-known result of Huber (1964) is that when $h$ is strongly unimodal, the least informative density $f_0 = (1 - \varepsilon)h + \varepsilon g_0$ has exponential tails. We study the minimax variance solutions when the known density $h$ is not necessarily strongly unimodal, and definitive results are obtained under mild regularity conditions on $h$. Examples are given where the support of the least informative contaminating density $g_0$ is a set of form: (i) $(-b, -a) \cup (a, b)$ for some $0 < a < b < \infty$; (ii) $(- < a < \infty;$ and (iii) a countable collection of disjoint sets. Minimax variance problems for multivariate location and scale parameters are also studied, with examples given of least informative distributions that are substochastic.