Multivariate spatial conditional U-quantiles: a Bahadur–Kiefer representation

Quantiles constitute a core concept in probability theory and theoretical statistics, providing an indispensable instrument in a wide array of applications. Although the univariate notion of quantiles is intuitively clear and mathematically well established, extending this concept to a multivariate framework poses significant theoretical and practical challenges. A well-established approach to extending univariate quantiles to the multivariate setting is the spatial (or geometric) framework, whose empirical counterparts exhibit notable robustness and admit an elegant Bahadur–Kiefer representation. Independently, another generalization of univariate quantiles leads to $U$-quantiles, which naturally encompass classical estimators such as the Hodges–Lehmann estimator for a central tendency. In this study, we bridge these perspectives by introducing multivariate conditional spatial $U$-quantiles and deriving their corresponding Bahadur–Kiefer representation. This representation enables us to establish fundamental theoretical properties, including weak convergence and a law of the iterated logarithm. These results are proved under some standard structural conditions on the Vapnik–Chervonenkis classes of functions and some mild conditions on the model. The uniform limit theorems discussed in this paper are key tools for further developments in data analysis involving empirical process techniques.