Deriving the asymptotic distribution of U- and V-statistics of dependent data using weighted empirical processes

Abstract

It is commonly acknowledged that V-functionals with an unbounded kernel are not Hadamard differentiable and that therefore the asymptotic distribution of U- and V-statistics with an unbounded kernel cannot be derived by the Functional Delta Method (FDM). However, in this article we show that V-functionals are quasi-Hadamard differentiable and that therefore a modified version of the FDM (introduced recently in (J. Multivariate Anal. 101 (2010) 2452–2463)) can be applied to this problem. The modified FDM requires weak convergence of a weighted version of the underlying empirical process. The latter is not problematic since there exist several results on weighted empirical processes in the literature; see, for example, (J. Econometrics 130 (2006) 307–335, Ann. Probab. 24 (1996) 2098–2127, Empirical Processes with Applications to Statistics (1986) Wiley, Statist. Sinica 18 (2008) 313–333). The modified FDM approach has the advantage that it is very flexible w.r.t. both the underlying data and the estimator of the unknown distribution function. Both will be demonstrated by various examples. In particular, we will show that our FDM approach covers mainly all the results known in literature for the asymptotic distribution of U- and V-statistics based on dependent data – and our assumptions are by tendency even weaker. Moreover, using our FDM approach we extend these results to dependence concepts that are not covered by the existing literature.