Yohai, Victor J.; Zamar, Ruben H. A minimax-bias property of the least \(\alpha\)-quantile estimates. (English) Zbl 0797.62027 Ann. Stat. 21, No. 4, 1824-1842 (1993). 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. Reviewer: J.A.Víšek (Praha) Cited in 2 ReviewsCited in 16 Documents MSC: 62F35 Robustness and adaptive procedures (parametric inference) 62J05 Linear regression; mixed models Keywords:minimax bias; robust estimates; \(M\)-functionals; \(S\)-functionals; \(LMS\)- functionals; \(R\)-functionals; least alpha-quantile estimating functionals; regression coefficients; absolute values of residuals; new notion of residual admissible estimating functional; leverage points; projection functionals; residual admissible × Cite Format Result Cite Review PDF Full Text: DOI