The Annals of Statistics

Automatic bandwidth choice and confidence intervals in nonparametric regression

Michael H. Neumann

Abstract

In the present paper we combine the issues of bandwidth choice and construction of confidence intervals in nonparametric regression. Main emphasis is put on fully data-driven methods. We modify the $\sqrt{n}$-consistent bandwidth selector of Härdle, Hall and Marron such that it is appropriate for heteroscedastic data, and we show how one can optimally choose the bandwidth g of the pilot estimator $\hat{m}_g$. Then we consider classical confidence intervals based on kernel estimators with data-driven bandwidths and compare their coverage accuracy. We propose a method to put undersmoothing with a data-driven bandwidth into practice and show that this procedure outperforms explicit bias correction.

Article information

Source
Ann. Statist., Volume 23, Number 6 (1995), 1937-1959.

Dates
First available in Project Euclid: 15 October 2002

https://projecteuclid.org/euclid.aos/1034713641

Digital Object Identifier
doi:10.1214/aos/1034713641

Mathematical Reviews number (MathSciNet)
MR1389859

Zentralblatt MATH identifier
0856.62042

Subjects
Primary: 62G15: Tolerance and confidence regions
Secondary: 62G07: Density estimation 62G20: Asymptotic properties

Citation

Neumann, Michael H. Automatic bandwidth choice and confidence intervals in nonparametric regression. Ann. Statist. 23 (1995), no. 6, 1937--1959. doi:10.1214/aos/1034713641. https://projecteuclid.org/euclid.aos/1034713641

References

• BERAN, R. 1986. Discussion of Jackknife, bootstrap and other resampling methods in nonparametric regression analysis,'' by C. F. J. Wu. Ann. Statist. 14 1295 1298. Z.
• FARAWAY, J. 1990. Bootstrap selection of bandwidth and confidence bands for nonparametric regression. J. Statist. Comput. Simulation 37 37 44. Z.
• FARAWAY, J. and JHUN, M. 1990. Bootstrap choice of bandwidth for density estimation. J. Amer. Statist. Assoc. 85 1119 1122. Z.
• GASSER, T. and MULLER, H. G. 1979. Kernel estimation of regression functions. Smoothing ¨ Techniques for Curve Estimation. Lecture Notes in Math. 757 23 68. Springer, New York. Z.
• GASSER, T., MULLER, H.-G. and MAMMITZSCH, V. 1985. Kernels for nonparametric curve estima¨ tion. J. Roy. Statist. Soc. Ser. B 47 238 252.
• HALL, P. 1991. Edgeworth expansions for nonparametric density estimators, with applications. Statistics 22 215 232. Z.
• HALL, P. 1992a. Effect of bias estimation on coverage accuracy of bootstrap confidence intervals for a probability density. Ann. Statist. 20 675 694. Z.
• HALL, P. 1992b. On bootstrap confidence intervals in nonparametric regression. Ann. Statist. 20 695 711. Z.
• HARDLE, W. and BOWMAN, A. 1988. Bootstrapping in nonparametric regression: local adaptive ¨ smoothing and confidence bands. J. Amer. Statist. Assoc. 83 102 110. Z.
• HARDLE, W., HALL, P. and MARRON, J. S. 1992. Regression smoothing parameters that are not ¨ far from their optimum. J. Amer. Statist. Assoc. 87 227 233. Z.
• HARDLE, W., HUET, S. and JOLIVET, E. 1995. Better bootstrap confidence intervals for regression ¨ curve estimation. Statistics. To appear. Z.
• HARDLE, W. and MAMMEN, E. 1993. Comparing nonparametric versus parametric regression ¨ fits. Ann. Statist. 21 1926 1947. Z.
• JENNRICH, R. I. 1969. Asy mptotic properties of nonlinear least squares estimators. Ann. Math. Statist. 40 633 643. Z.
• JONES, M. C. and SHEATHER, S. J. 1991. Using non-stochastic terms to advantage in kernel-based estimation of integrated squared density derivatives. Statist. Probab. Lett. 11 511 514. Z.
• MULLER, H. G. 1988. Nonparametric Regression Analy sis of Longitudinal Data. Lecture Notes ¨ in Statist. 46. Springer, Berlin. Z.
• MULLER, H. G. and STADTMULLER, U. 1987. Variable bandwidth kernel estimators of regression ¨ ¨ curves. Ann. Statist. 15 182 201. Z.
• NEUMANN, M. H. 1992a. On completely data-driven pointwise confidence intervals in nonparametric regression. Rapport Technique 92-02, INRA, Dept. Biometrie, Jouy-en-Josas, ´ ´ France. Z.
• NEUMANN, M. H. 1992b. Pointwise confidence intervals in nonparametric regression with heteroscedastic error structure. Preprint No. 34, Institut fur Angewandte Analy sis ¨ und Stochastik, Berlin. Z.
• SHEATHER, S. J. and JONES, M. C. 1991. A reliable data-based bandwidth selection method for kernel density estimation. J. Roy. Statist. Soc. Ser. B 53 683 690. Z.
• SKOVGAARD, I. M. 1981. Transformations of an Edgeworth expansion by a sequence of smooth functions. Scand. J. Statist. 8 207 217. Z.
• SKOVGAARD, I. M. 1986. On multivariate Edgeworth expansions. Internat. Statist. Rev. 54 169 186. Z.
• WHITTLE, P. 1960. Bounds for the moments of linear and quadratic forms in independent variables. Theory Probab. Appl. 5 302 305. Z.
• WU, C. F. J. 1986. Jackknife, bootstrap and other resampling methods in nonparametric Z. regression analysis with discussion. Ann. Statist. 14 1261 1344.