Estimators Related to U-Processes with Applications to Multivariate Medians: Asymptotic Normality

If a criterion function g(x₁, …, xₘ; θ) depends on $m > 1$ samples, then a natural estimator of max Pᵐg(x₁, …, xₘ; θ) is the max of a U-process. It is observed that, under suitable conditions, these estimators are asymptotically normal. This is then applied to prove asymptotic normality of Liu's simplical median and of Oja's medians in Rᵈ.