We consider the distributional and the almost sure pointwise Bahadur-Kiefer representation for U-quantiles. We show that the order of this representation depends on the order of the local variance of the empirical process of U-statistic structure at the U-quantile. Our results indicate that U-quantiles can be smoother than quantiles. U-quantiles can either be as unsmooth as quantiles or can behave as differentiable statistical functionals.
"The Bahadur-Kiefer representation for U-quantiles." Ann. Statist. 24 (3) 1400 - 1422, June 1996. https://doi.org/10.1214/aos/1032526976