Bernoulli

  • Bernoulli
  • Volume 18, Number 3 (2012), 803-822.

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

Eric Beutner and Henryk Zähle

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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.

Article information

Source
Bernoulli, Volume 18, Number 3 (2012), 803-822.

Dates
First available in Project Euclid: 28 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.bj/1340887003

Digital Object Identifier
doi:10.3150/11-BEJ358

Mathematical Reviews number (MathSciNet)
MR2948902

Zentralblatt MATH identifier
06064463

Keywords
Functional Delta Method Jordan decomposition quasi-Hadamard differentiability stationary sequence of random variables U- and V-statistic weak dependence weighted empirical process

Citation

Beutner, Eric; Zähle, Henryk. Deriving the asymptotic distribution of U- and V-statistics of dependent data using weighted empirical processes. Bernoulli 18 (2012), no. 3, 803--822. doi:10.3150/11-BEJ358. https://projecteuclid.org/euclid.bj/1340887003


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