The Annals of Statistics

Uniform in bandwidth consistency of kernel-type function estimators

Uwe Einmahl and David M. Mason

Full-text: Open access

Abstract

We introduce a general method to prove uniform in bandwidth consistency of kernel-type function estimators. Examples include the kernel density estimator, the Nadaraya–Watson regression estimator and the conditional empirical process. Our results may be useful to establish uniform consistency of data-driven bandwidth kernel-type function estimators.

Article information

Source
Ann. Statist., Volume 33, Number 3 (2005), 1380-1403.

Dates
First available in Project Euclid: 1 July 2005

Permanent link to this document
https://projecteuclid.org/euclid.aos/1120224106

Digital Object Identifier
doi:10.1214/009053605000000129

Mathematical Reviews number (MathSciNet)
MR2195639

Zentralblatt MATH identifier
1079.62040

Subjects
Primary: 60F15: Strong theorems 62G07: Density estimation 62G08: Nonparametric regression

Keywords
Kernel-type function estimator uniform in bandwidth consistency

Citation

Einmahl, Uwe; Mason, David M. Uniform in bandwidth consistency of kernel-type function estimators. Ann. Statist. 33 (2005), no. 3, 1380--1403. doi:10.1214/009053605000000129. https://projecteuclid.org/euclid.aos/1120224106


Export citation

References

  • Bosq, D. and Lecoutre, J.-P. (1987). Théorie de l'estimation fonctionnelle. Economica, Paris.
  • Breiman, L., Meisel, W. and Purcell, E. (1977). Variable kernel estimate of multivariate densities. Technometrics 19 135–144.
  • Collomb, G. (1979). Conditions nécessaires et suffisantes de convergence uniforme d'un estimateur de régression, estimation des dérivées de la régression. C. R. Acad. Sci. Paris Sér. A 288 161–163.
  • Deheuvels, P. (1974). Conditions nécessaires et suffisantes de convergence ponctuelle presque sûre et uniforme presque sûre des estimateurs de la densité. C. R. Acad. Sci. Paris Sér. A 278 1217–1220.
  • Deheuvels, P. (2000). Uniform limit laws for kernel density estimators on possibly unbounded intervals. In Recent Advances in Reliability Theory: Methodology, Practice and Inference (N. Limnios and M. Nikulin, eds.) 477–492. Birkhäuser, Basel.
  • Deheuvels, P. and Mason, D. M. (1992). Functional laws of the iterated logarithm for the increments of empirical and quantile processes. Ann. Probab. 20 1248–1287.
  • Deheuvels, P. and Mason, D. M. (2004). General asymptotic confidence bands based on kernel-type function estimators. Stat. Inference Stoch. Process. 7 225–277.
  • Devroye, L. and Györfi, L. (1985). Nonparametric Density Estimation: The $L_1$ View. Wiley, New York.
  • Devroye, L. and Lugosi, G. (2001). Combinatorial Methods in Density Estimation. Springer, New York.
  • Eggermont, P. P. B. and LaRiccia, V. N. (2001). Maximum Penalized Likelihood Estimation 1. Density Estimation. Springer, New York.
  • Einmahl, U. and Mason, D. M. (2000). An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theoret. Probab. 13 1–37.
  • Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Chapman and Hall, London.
  • Giné, E. and Guillou, A. (2001). On consistency of kernel density estimators for randomly censored data: Rates holding uniformly over adaptive intervals. Ann. Inst. H. Poincaré Probab. Statist. 37 503–522.
  • Giné, E. and Guillou, A. (2002). Rates of strong uniform consistency for multivariate kernel density estimators. Ann. Inst. H. Poincaré Probab. Statist. 38 907–921.
  • Giné, E. and Zinn, J. (1984). Some limit theorems for empirical processes (with discussion). Ann. Probab. 12 929–998.
  • Härdle, W., Janssen, P. and Serfling, R. (1988). Strong uniform consistency rates for estimators of conditional functionals. Ann. Statist. 16 1428–1449.
  • Hall, P. (1992). On global properties of variable bandwidth density estimators. Ann. Statist. 20 762–778.
  • Jones, M. C., Marron, J. S. and Sheather, S. J. (1996). A brief survey of bandwidth selection for density estimation. J. Amer. Statist. Assoc. 91 401–407.
  • Ledoux, M. (1996). On Talagrand's deviation inequalities for product measures. ESAIM Probab. Statist. 1 63–87.
  • Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Springer, Berlin.
  • Lepski, O. V., Mammen, E. and Spokoiny, V. G. (1997). Optimal spatial adaptation to inhomogenous smoothness: An approach based on kernel estimates with variable bandwidth selectors. Ann. Statist. 25 929–947.
  • Lepski, O. V. and Spokoiny, V. G. (1997). Optimal pointwise adaptive methods in nonparametric estimation. Ann. Statist. 25 2512–2546.
  • Müller, H.-G. and Stadtmüller, U. (1987). Variable bandwidth kernel estimators of regression curves. Ann. Statist. 15 182–201.
  • Nolan, D. and Marron, J. S. (1989). Uniform consistency of automatic and location-adaptive delta-sequence estimators. Probab. Theory Related Fields 80 619–632.
  • Nolan, D. and Pollard, D. (1987). $U$-processes: Rates of convergence. Ann. Statist. 15 780–799.
  • Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.
  • Staniswalis, J. G. (1989). Local bandwidth selection for kernel estimates. J. Amer. Statist. Assoc. 84 284–288.
  • Stein, E. M. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press.
  • Stute, W. (1982). The oscillation behavior of empirical processes. Ann. Probab. 10 86–107.
  • Stute, W. (1982). The law of the iterated logarithm for kernel density estimators. Ann. Probab. 10 414–422.
  • Stute, W. (1984). The oscillation behavior of empirical processes: The multivariate case. Ann. Probab. 12 361–379.
  • Stute, W. (1986). On almost sure convergence of conditional empirical distribution functions. Ann. Probab. 14 891–901.
  • Talagrand, M. (1994). Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22 28–76.
  • van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Univ. Press.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.