M-estimation, convexity and quantiles

• 成果类型: Article
• 署名作者: Koltchinskii, VI
• 署名单位: University of New Mexico
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1031833659
• 发表日期: 1997
• 页码: 435-477
• 关键词: regression quantiles; multivariate data; uniqueness; symmetry

摘要

The paper develops a class of extensions of the univariate quantile function to the multivariate case (M-quantiles), related in a certain way to M-parameters of a probability distribution and their M-estimators. The spatial (geometric) quantiles, recently introduced by Koltchinskii and Dudley and by Chaudhuri as well as the regression quantiles of Koenker and Basset, are the examples of the M-quantile function discussed in the paper. We study the main properties of M-quantiles and develop the asymptotic theory of empirical M-quantiles. We use M-quantiles to extend L-parameters and L-estimators to the multivariate case; to introduce a bootstrap test for spherical symmetry of a multivariate distribution, and to extend the notion of regression quantiles to multiresponse linear regression models.