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

Wiens, Doug

Ann. Statist., Tome 14 (1986) no. 2, p. 724-732 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 1986-06-14
Classification:  Robust estimation,  $M$-estimators,  minimax variance,  Kolmogorov neighbourhood,  minimum Fisher information,  62G35,  62G05