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$$M$$-estimation, convexity and quantiles. (English) Zbl 0878.62037
This paper develops a class of extensions of univariate quantile functions to the multivariate case, related in a certain way to $$M$$-parameters of a probability distribution and their $$M$$-estimators. An $$M$$-parameter with respect to a distribution $$P$$ and some integrable function $$f(s,.)$$, $$s\in\mathbb{R}^d$$, is a minimal point $$s_0(P)$$ of the functional $$\int f(s,x)P(dx)$$ and the associated $$M$$-estimator is defined to be the corresponding $$M$$-parameter when $$P$$ is replaced by the empirical distribution $$\widehat{P}_n= n^{-1}(\delta_{X_1}+\cdots+ \delta_{X_n})$$. The spatial (geometric) quantiles, recently introduced by P. Chaudhuri [J. Am. Stat. Assoc. 91, No. 434, 862-872 (1996; Zbl 0869.62040)] as well as the so-called regression quantiles of R. Koenker and G. Bassett [Econometrica 46, 33-50 (1978; Zbl 0373.62038)] are special cases of the $$M$$-quantile function which is the subject of the paper.
The main properties of the $$M$$-quantiles are studied and an asymptotic theory of empirical $$M$$-quantiles is developed. Furthermore, $$M$$-quantiles are used to extend $$L$$-parameters and $$L$$-estimators to the multivariate case in order to construct a bootrap test for spherical symmetry of a multivariate distribution and to extend the notion of regression quantiles to multiresponse linear regression models.

##### MSC:
 62G20 Asymptotic properties of nonparametric inference 62G30 Order statistics; empirical distribution functions 60F05 Central limit and other weak theorems 62E20 Asymptotic distribution theory in statistics 60F17 Functional limit theorems; invariance principles
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