Minimax Properties of $M-, R-$ and $L$-Estimators of Location in Levy Neighbourhoods

Abstract

In the context of Huber's theory of robust estimation of a location parameter, the literature on minimax properties of $M$-, $R$- and $L$-estimators is surveyed. New results are obtained for the model in which the unknown error distribution is assumed to lie in a Levy neighbourhood of a symmetric distribution $G: \mathscr{P}_{\varepsilon,\delta}(G) = \{F\mid G(x - \delta) - \varepsilon \leq F(x) \leq G(x + \delta) + \varepsilon \text{for all} x\}$. Under reasonably general conditions on $G$, the distribution $F_0$ in $\mathscr{P}_{\varepsilon,\delta}(G)$ which minimizes Fisher information for location is found. Huber's minimax property for $M$-estimators is shown to hold for $R$-estimators but to fail for $L$-estimators in Levy neighbourhoods. The latter is proved by constructing a subneighbourhood of distributions $\mathscr{F}_0$, with $F_0 \in \mathscr{F}_0 \subset \mathscr{P}_{\varepsilon\delta}(G)$, such that the asymptotic variance of the $L$-estimator which is asymptotically efficient at $F_0$ is minimized over $\mathscr{F}_0$ at $F_0$.