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A depth-based rank sum statistic for multivariate data introduced by Liu and Singh [J. Amer. Statist. Assoc. 88 (1993) 252–260] as an extension of the Wilcoxon rank sum statistic for univariate data has been used in multivariate rank tests in quality control and in experimental studies. Those applications, however, are based on a conjectured limiting distribution, provided by Liu and Singh [J. Amer. Statist. Assoc. 88 (1993) 252–260]. The present paper proves the conjecture…

基于投影深度的多维截尾均值AI 解读

As estimators of location parameters, univariate trimmed means are well known for their robustness and efficiency. They can serve as robust alternatives to the sample mean while possessing high efficiencies at normal as well as heavy-tailed models. This paper introduces multidimensional trimmed means based on projection depth induced regions. Robustness of these depth trimmed means is investigated in terms of the influence function and finite sample breakdown point. The…

深度加权散布矩阵估计量AI 解读

General depth weighted scatter estimators are introduced and investigated. For general depth functions, we find out that these affine equivariant scatter estimators are Fisher consistent and unbiased for a wide range of multivariate distributions, and show that the sample scatter estimators are strong and $\sqrt{n}$-consistent and asymptotically normal, and the influence functions of the estimators exist and are bounded in general. We then concentrate on a specific case of…

投影深度估计量的影响函数与最大偏差AI 解读

Location estimators induced from depth functions increasingly have been pursued and studied in the literature. Among them are those induced from projection depth functions. These projection depth based estimators have favorable properties among their competitors. In particular, they possess the best possible finite sample breakdown point robustness. However, robustness of estimators cannot be revealed by the finite sample breakdown point alone. The influence function, gross…

Stahel–Donoho 估计量与多元数据的深度加权均值AI 解读

The depth of multivariate data can be used to construct weighted means as robust estimators of location. The use of projection depth leads to the Stahel-Donoho estimator as a special case. In contrast to maximal depth estimators, the depth-weighted means are shown to be asymptotically normal under appropriate conditions met by depth functions commonly used in the current literature. We also confirm through a finite-sample study that the Stahel-Donoho estimator achieves a…

基于投影的深度函数及其诱导中位数AI 解读

A class of projection-based depth functions is introduced and studied. These projection-based depth functions possess desirable properties of statistical depth functions and their sample versions possess strong and order $\sqrt{n}$ uniform consistency. Depth regions and contours induced from projection-based depth functions are investigated. Structural properties of depth regions and contours and general continuity and convergence results of sample depth regions are…