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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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