Minimax variance M-estimators of location in Kolmogorov neighbourhoods
type: Item
instance of: scholarly article
title: Minimax variance M-estimators of location in Kolmogorov neighbourhoods
zbMATH Open document ID: 0603.62043
author: Doug Wiens
published in: The Annals of Statistics
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 62F35
Mathematics Subject Classification ID: 62G05
zbMATH DE Number: 3974072
zbMATH Keywords: minimax variance
zbMATH Keywords: minimum Fisher information
zbMATH Keywords: location
zbMATH Keywords: Kolmogorov neighbourhoods
zbMATH Keywords: M-estimators
zbMATH Keywords: most robust
zbMATH Keywords: normal distribution
zbMATH Keywords: strongly unimodal densities
zbMATH Keywords: Student's t
zbMATH Keywords: nonmonotonic score functions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176349949
OpenAlex ID: W2091624215
DOI: 10.1214/AOS/1176349949
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description: scientific article; zbMATH DE number 3974072