Minimax variance M-estimators of location in Kolmogorov neighbourhoods
The author considers those distributions with minimum Fisher information for location in various Kolmogorov neighbourhoods \[ K_{\epsilon}=\{F| \sup_{x}| F(x)-G(x)| \leq \epsilon \} \] 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. In the case \(G=\Phi\), the normal distribution, \textit{P. J. Huber} [Ann. Math. Stat. 35, 73-101 (1964; Zbl 0136.398)] obtained the most robust M-estimator for \(\epsilon\leq 0.0303\), \textit{J. Sacks} and \textit{D. Ylvisaker} [ibid. 43, 1068-1075 (1972; Zbl 0284.62022)] for \(\epsilon\geq 0.0303.\) The solution of Huber is shown to apply to all distributions with strongly unimodal densities, whose score functions satisfy a further condition, and the solution of Sacks and Ylvisaker is shown to apply under much weaker conditions. New forms of the solution are given for such distribution as Student's t, with nonmonotonic score functions. The general form of the solution is discussed.
scientific article; zbMATH DE number 3974072