A minimax-bias property of the least \(\alpha\)-quantile estimates
type: Item
instance of: scholarly article
title: A minimax-bias property of the least \(\alpha\)-quantile estimates
zbMATH Open document ID: 0797.62027
author: Ruben H. Zamar
author: Victor J. Yohai
published in: The Annals of Statistics
publication date: 1994-10-25T00:00:00Z
review text: The least \(\alpha\)-quantile estimating functional \(T\) is introduced as a vector of regression coefficients such that the \(\alpha\)-quantile of the distribution function of absolute values of residuals (d.f.a.v.r.), with respect to the underlying probability measure, is minimal. Also a new notion of residual admissible estimating functional (r.a.e.f.) is defined so that it implies such a d.f.a.v.r. that for any fixed vector of regression coefficients we cannot obtain a d.f.a.v.r. larger at all the points of the positive halfaxis. Then it is shown that the least \(\alpha\)-quantile regression functionals are r.a.e.f. The same is proved for \(M\)-, \(S\)-, \(LMS\)-, \(LTS\)- and some \(R\)-functionals. It is also shown that the estimating functionals penalizing high leverage points, as \(GM\)- or projection functionals, are not residual admissible.
Mathematics Subject Classification ID: 62F35
Mathematics Subject Classification ID: 62J05
zbMATH DE Number: 528649
zbMATH Keywords: minimax bias
zbMATH Keywords: robust estimates
zbMATH Keywords: \(M\)-functionals
zbMATH Keywords: \(S\)-functionals
zbMATH Keywords: \(LMS\)- functionals
zbMATH Keywords: \(R\)-functionals
zbMATH Keywords: least alpha-quantile estimating functionals
zbMATH Keywords: regression coefficients
zbMATH Keywords: absolute values of residuals
zbMATH Keywords: new notion of residual admissible estimating functional
zbMATH Keywords: leverage points
zbMATH Keywords: projection functionals
zbMATH Keywords: residual admissible
reviewed by: Jan Ámos Víšek
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176349400
OpenAlex ID: W2022370612
DOI: 10.1214/AOS/1176349400
description: scientific article; zbMATH DE number 528649