Minimax Variance $M$-Estimators of Location in Kolmogorov Neighbourhoods

Abstract

We exhibit those distributions with minimum Fisher information for location in various Kolmogorov neighbourhoods $\{F|\sup_x|F(x) - G(x)| \leq \varepsilon\}$ of a fixed, symmetric distribution $G$. The associated $M$-estimators are then most robust (in Huber's minimax sense) for location estimation within these neighbourhoods. The previously obtained solution of Huber (1964) for $G = \Phi$ and "small" $\varepsilon$ is shown to apply to all distributions with strongly unimodal densities whose score functions satisfy a further condition. The "large" $\varepsilon$ solution for $G = \Phi$ of Sacks and Ylvisaker (1972) is shown to apply under much weaker conditions. New forms of the solution are given for such distributions as "Student's" $t$, with nonmonotonic score functions. The general form of the solution is discussed.