The ``automatic'' robustness of minimum distance functionals
type: Item
instance of: scholarly article
title: The ``automatic'' robustness of minimum distance functionals
zbMATH Open document ID: 0684.62030
author: David Donoho
author: Richard C. Liu
published in: The Annals of Statistics
publication date: 1988-01-01T00:00:00Z
review text: If \(\{P_{\theta}\}\) is a model indexed by real parameter \(\theta\) and \(\mu\) is a metric between probabilities then the minimum distance (MD) functional \({\hat \theta}\)(P) is a solution of \(\mu (P,P_{{\hat \theta}})=\min_{\theta}\mu (P,P_{\theta}).\) The authors show that the functional is very stable with respect to the quantity being estimated in the following sense. Denote by \(\delta\) a measure of distortion of an ideal distribution \(P_ 0\) and define the bias distortion curve for a functional T to be \[ b(\epsilon)=\sup \{| T(P)-T(P_ 0)|:\quad \delta (P,P_ 0)\leq \epsilon \}. \] The slope \(\gamma^*\) of b(\(\epsilon)\) at zero can be sometimes interpreted as the gross error sensitivity while \(\epsilon^*\) is equal to the smallest \(\epsilon\) such that the graph of b has no singularities in the ball of radius \(\leq \epsilon\) as the break down point of T. The basic proposition of the paper claims that whenever the distortion measure \(\delta =\mu\) then for the MD functionals based on \(\mu\) we have \(b_ 0(\epsilon)\leq b(\epsilon)\leq b_ 0(2\epsilon),\) where \[ b_ 0(\epsilon)=\sup \{| \theta -\theta_ 0|:\quad \mu (P_{\theta},P_ 0)\}. \] This then implies \(\gamma^*(\theta)<2 \inf_{T} \gamma^*(T)\) and \(\epsilon^*(\theta)>\sup_{T} \epsilon^*(T)/2\), where the inf and sup are taken over Fisher consistent T. Thus the MD functional is shown to have within a factor of 2 the smallest sensitivity and the best breakdown point among Fisher consistent functionals. These statements are then improved for Hilbertian distances and invariant cases. Generalizations are given for multidimensional parameter models, semiparametric models and minimum discrepancy functionals. Also consistency and finite sample robustness of MD estimators are studied.
Mathematics Subject Classification ID: 62F35
Mathematics Subject Classification ID: 62F12
zbMATH DE Number: 4121220
zbMATH Keywords: minimum distance functional
zbMATH Keywords: quantitative robustness
zbMATH Keywords: gross-error sensitivity
zbMATH Keywords: breakdown point
zbMATH Keywords: Kolmogorov
zbMATH Keywords: Lévy
zbMATH Keywords: Prokhorov
zbMATH Keywords: variation
zbMATH Keywords: Hellinger distances
zbMATH Keywords: Cramér-von Mises discrepancy
zbMATH Keywords: bias distortion curve
zbMATH Keywords: break down point
zbMATH Keywords: distortion measure
zbMATH Keywords: Fisher consistent functionals
zbMATH Keywords: multidimensional parameter models
zbMATH Keywords: semiparametric models
zbMATH Keywords: minimum discrepancy functionals
zbMATH Keywords: consistency
zbMATH Keywords: finite sample robustness
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176350820
OpenAlex ID: W1968956560
DOI: 10.1214/AOS/1176350820
description: scientific article; zbMATH DE number 4121220