Kybernetika 47 no. 6, 855-865, 2011

One Bootstrap suffices to generate sharp uniform Bounds in Functional Estimation

Paul Deheuvels


We consider, in the framework of multidimensional observations, nonparametric functional estimators, which include, as special cases, the Akaike-Parzen-Rosenblatt kernel density estimators ([1], [18], [20]), and the Nadaraya-Watson kernel regression estimators ([16], [22]). We evaluate the sup-norm, over a given set <span class="tex"><b>I</b></span>, of the difference between the estimator and a non-random functional centering factor (which reduces to the estimator mean for kernel density estimation). We show that, under suitable general conditions, this random quantity is consistently estimated by the sup-norm over <span class="tex"><b>I</b></span> of the difference between the original estimator and a bootstrapped version of this estimator. This provides a simple and flexible way to evaluate the estimator accuracy, through a single bootstrap. The present work generalizes former results of Deheuvels and Derzko [4], given in the setup of density estimation in <b>R</b>.


density estimation, nonparametric functional estimation, regression estimation, bootstrap, resampling methods, confidence regions, empirical processes


62G05, 62G08, 62G09, 62G15, 62G20, 62G30


  1. H. Akaike: An approximation of the density function. Ann. Inst. Statist. Math. 6 (1954), 127-132.   CrossRef
  2. P. J. Bickel and M. Rosenblatt: On some global measures of the deviations of density functions estimates. Ann. Statist. 1 (1973), 1071-1095.   CrossRef
  3. G. Claeskens and I. van Keilegom: Bootstrap confidence bands for regression curves and their derivatives. Ann. Statist. 31 (2003), 1852-1884.   CrossRef
  4. P. Deheuvels and G. Derzko: Asymptotic certainty bands for kernel density estimators based upon a bootstrap resampling scheme. In: Statistical Models and Methods for Biomedical and Technical systems, Stat. Ind. Technol., Birkhäuser, Boston MA 2008, pp. 171-186.   CrossRef
  5. P. Deheuvels and D. M. Mason: General asymptotic confidence bands based on kernel-type function estimators. Statist. Infer. Stoch. Processes 7 (2004), 225-277.   CrossRef
  6. R. M. Dudley: Uniform Central Limit Theorems. Cambridge University Press, Cambridge 1999.   CrossRef
  7. B. Efron: Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 (1979), 1-26.   CrossRef
  8. U. Einmahl and D. M. Mason: Uniform in bandwidth consistency of kernel-type function estimators. Ann. Statist. 33 (2005), 1380-1403.   CrossRef
  9. E. Giné, V. Koltchinskii and L. Sakhanenko: Kernel density estimators: convergence in distribution for weighted sup-norms. Probab. Theory Related Fields 130 (2004), 167-198.   CrossRef
  10. E. Giné and D. M. Mason: On local $U$-statistic processes and the estimation of densities of functions of several sample variables. Ann. Statist. 35 (2007), 1105-1145.   CrossRef
  11. P. Hall: The Bootstrap and Edgeworth Expansion. Springer, New York 1992.   CrossRef
  12. W. Härdle and J. S. Marron: Bootstrap simultaneous error bars for nonparametric regression. Ann. Statist. 19 (1991), 778-796.   CrossRef
  13. W. Härdle and A. W. Bowman: Bootstrapping in nonparametric regression: Local adaptative smoothing and confidence bands. J. Amer. Statist. Assoc. 83 (1988), 102-110.   CrossRef
  14. G. Li and S. Datta: A bootstrap approach to nonparametric regression for right censored data. Ann. Inst. Statist. Math. 53 (2001), 708-729.   CrossRef
  15. D. M. Mason and M. A. Newton: A rank statistics approach to the consistency of a general bootstrap. Ann. Statist. 20 (1992), 1611-1624.   CrossRef
  16. E. A. Nadaraya: On estimating regression. Theor. Probab. Appl. 9 (1964), 141-142.   CrossRef
  17. D. Nolan and D. Pollard: $U$-processes: Rates of convergence. Ann. Statist. 15 (1987), 780-799.   CrossRef
  18. E. Parzen: On the estimation of probability density and mode. Ann. Math. Statist. 33 (1962), 1065-1076.   CrossRef
  19. D. Pollard: Convergence of Stochastic Processes. Springer, New York 1984.   CrossRef
  20. M. Rosenblatt: Remarks on some nonparametric estimates of a density function. Ann. Math. Statist. 27 (1956), 832-837.   CrossRef
  21. A. W. van der Vaart and J. A. Wellner: Weak Convergence and Empirical Processes with Aplications to Statistics. Springer, New York 1996.   CrossRef
  22. G. S. Watson: Smooth regression analysis. Sankhy\={a} A 26 (1964), 359-372.   CrossRef