Multivariate Spatial U-Quantiles: a Bahadur-Kiefer Representation, a Theil-Sen Estimator for Multiple Regression, and a Robust Dispersion Estimator
نویسندگان
چکیده
A leading multivariate extension of the univariate quantiles is the so-called “spatial” or “geometric” notion, for which sample versions are highly robust and conveniently satisfy a Bahadur-Kiefer representation. Another extension of univariate quantiles has been to univariate U-quantiles, on the basis of which, for example, the well-known Hodges-Lehmann location estimator has a natural formulation. Generalizing both extensions, we introduce multivariate spatial U-quantiles and develop a corresponding Bahadur-Kiefer representation. New statistics based on spatial U-quantiles are presented for nonparametric estimation of multiple regression coefficients, extending the classical Theil-Sen nonparametric simple linear regression slope estimator, and for robust estimation of multivariate dispersion. Some other applications are mentioned as well. AMS 2000 Subject Classification: Primary 62G99 Secondary 60F15.
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