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April 1997 M-estimation, convexity and quantiles
V. I. Koltchinskii
Ann. Statist. 25(2): 435-477 (April 1997). DOI: 10.1214/aos/1031833659


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 useM-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.


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V. I. Koltchinskii. "M-estimation, convexity and quantiles." Ann. Statist. 25 (2) 435 - 477, April 1997.


Published: April 1997
First available in Project Euclid: 12 September 2002

zbMATH: 0878.62037
MathSciNet: MR1439309
Digital Object Identifier: 10.1214/aos/1031833659

Primary: 60F05
Secondary: 60F17 , 62E20

Keywords: $M$-estimator , $M$-parameter , Empirical processes , empirical quantiles , quantile function , regression quantiles , spatial quantiles

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 2 • April 1997
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