Bernoulli

  • Bernoulli
  • Volume 14, Number 4 (2008), 1108-1133.

Uniform in bandwidth consistency of conditional $U$-statistics

Julia Dony and David M. Mason

Full-text: Open access

Abstract

Stute [Ann. Probab. 19 (1991) 812–825] introduced a class of estimators called conditional $U$-statistics. They can be seen as a generalization of the Nadaraya–Watson estimator for the regression function. Stute proved their strong pointwise consistency to $$m(\mathbf{t}):=\mathbb{E}[g(Y_{1},\ldots,Y_{m})|(X_{1},\ldots,X_{m})=\mathbf{t}],\qquad\mathbf{t}\in\mathbb{R}^{m}.$$ Very recently, Giné and Mason introduced the notion of a local $U$-process, which generalizes that of a local empirical process, and obtained central limit theorems and laws of the iterated logarithm for this class. We apply the methods developed in Einmahl and Mason [Ann. Statist. 33 (2005) 1380–1403] and Giné and Mason [Ann. Statist. 35 (2007) 1105–1145; J. Theor. Probab. 20 (2007) 457–485] to establish uniform in t and in bandwidth consistency to m(t) of the estimator proposed by Stute. We also discuss how our results are used in the analysis of estimators with data-dependent bandwidths.

Article information

Source
Bernoulli, Volume 14, Number 4 (2008), 1108-1133.

Dates
First available in Project Euclid: 6 November 2008

Permanent link to this document
https://projecteuclid.org/euclid.bj/1225980573

Digital Object Identifier
doi:10.3150/08-BEJ136

Mathematical Reviews number (MathSciNet)
MR2543588

Zentralblatt MATH identifier
1169.62037

Keywords
conditional $U$-statistics consistency data-dependent bandwidth selection empirical process kernel estimation Nadaraya–Watson regression uniform in bandwidth

Citation

Dony, Julia; Mason, David M. Uniform in bandwidth consistency of conditional $U$-statistics. Bernoulli 14 (2008), no. 4, 1108--1133. doi:10.3150/08-BEJ136. https://projecteuclid.org/euclid.bj/1225980573


Export citation

References

  • [1] Brown, B. (1971). Martingale central limit theorems., Ann. Math. Statist. 42 59–66.
  • [2] de la Peña, V.H. and Giné, E. (1999)., Decoupling. From Dependence to Independence. Randomly Stopped Processes. U-Statistics and Processes. Martingales and Beyond.Probability and Its Applications. New York: Springer.
  • [3] Dony, J., Einmahl, U. and Mason, D.M. (2006). Uniform in bandwidth consistency of local polynomial regression function estimators., Austr. J. Stat. 35 105–120.
  • [4] Einmahl, U. and Mason, D.M. (2000). An empirical process approach to the uniform consistency of kernel-type function estimators., J. Theor. Probab. 13 1–37.
  • [5] Einmahl, U. and Mason, D.M. (2005). Uniform in bandwidth consistency of kernel-type function estimators., Ann. Statist. 33 1380–1403.
  • [6] Giné, E. and Mason, D.M. (2007a). On local, U-statistic processes and the estimation of densities of functions of several sample variables. Ann. Statist. 35 1105–1145.
  • [7] Giné, E. and Mason, D.M. (2007b). Laws of the iterated logarithm for the local, U-statistic process. J. Theor. Probab. 20 457–485.
  • [8] Härdle, W. and Marron, J.S. (1985). Optimal bandwidth selection in nonparametric regression function estimation., Ann. Statist. 13 1465–1481.
  • [9] Hall, P. (1984). Asymptotic properties of integrated square error and cross-validation for kernel estimation of a regression function., Z. Wahrsch. Verw. Gebiete 67 175–196.
  • [10] Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution., Ann. Math. Statist. 19 293–325.
  • [11] Rachdi, M. and Vieu, P. (2007). Nonparametric regression for functional data: Automatic smoothing parameter selection., J. Statist. Plann. Inference 137 2784–2801.
  • [12] Sen, A. (1994). Uniform strong consistency rates for conditional, U-statistics. Sankhyā 56 Ser. A 179–194.
  • [13] Stute, W. (1991). Conditional, U-statistics. Ann. Probab. 19 812–825.
  • [14] Talagrand, M. (1994). Sharper bounds for Gaussian and empirical processes., Ann. Probab. 22 28–76.
  • [15] Tsybakov, A.B. (1987). On the choice of bandwidth in nonparametric kernel regression., Teor. Veroyatnost. i Primenen. 32 153–159. (In Russian.)
  • [16] van der Vaart, A.W. and Wellner, J.A. (1996)., Weak Convergence and Empirical Processes with Applications to Statistics. New York: Springer.
  • [17] Vieu, P. (1991). Nonparametric regression: Optimal local bandwidth choice., J. Roy. Statist. Soc. Ser. B 53 453–464.