Uniform consistency and uniform in bandwidth consistency for nonparametric regression estimates and conditional U-statistics involving functional data

W. Stute [(1991), Annals of Probability, 19, 812-825] introduced a class of so-called conditional U-statistics, which may be viewed as a generalisation of the Nadaraya-Watson estimates of a regression function. Stute proved their strong pointwise consistency toWe apply the methods developed in Dony and Mason [(2008), Bernoulli, 14(4), 1108-1133] to establish uniformity in and in bandwidth consistency (i.e. , where at some specific rate) to of the estimator proposed by Stute when Y and covariates X are functional taking value in some abstract spaces. In addition, uniform consistency is also established over for a suitably restricted class . The theoretical uniform consistency results, established in this paper, are (or will be) key tools for many further developments in functional data analysis. Applications include the Nadaraya-Watson kernel estimators and the conditional distribution function. Our theorems allow data-driven local bandwidths for these statistics.