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.
Citation
John R. Collins. Douglas P. Wiens. "Minimax Variance M-Estimators in $\varepsilon$-Contamination Models." Ann. Statist. 13 (3) 1078 - 1096, September, 1985. https://doi.org/10.1214/aos/1176349657
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