A minimax-bias property of the least \(\alpha\)-quantile estimates
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.
least alpha-quantile estimating functionals
new notion of residual admissible estimating functional
scientific article; zbMATH DE number 528649