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Interval and band estimation for curves with jumps

Part of: Nonparametric inference

Published online by Cambridge University Press:  14 July 2016

Irène Gijbels ,
Peter Hall  and
Aloïs Kneip
Irène Gijbels
Affiliation:
Institut de Statistique, Université Catholique de Louvain, 20 Voie du Roman Pays, B-1348 Louvain-la-Neuve, Belgium. Email address: [email protected]
Peter Hall
Affiliation:
Centre for Mathematics and Its Applications, Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia. Email address: [email protected]
Aloïs Kneip
Affiliation:
Johannes Gutenberg Universität-Mainz, Lehrstuhl für Statistik und Mathematik FB, Rechts-und Witschaftswissenschaften, Jakob-Welder-Weg 9, D-55128 Mainz, Germany. Email address: [email protected]
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Abstract

Jump points in curves arise when the conditions under which data are generated change suddenly, for example because of an unplanned change in a treatment. This paper suggests bootstrap methods for quantifying the error in estimates of jump points, and for constructing confidence intervals for jump points and confidence bands for the curve. These problems have the unusual feature that the sampling error of the jump-point estimator often has a highly non-normal distribution, which depends intimately on the distribution of regression errors. The methods are illustrated by a simulation study as well as by an application to data on the annual flow volume of the Nile river.

Keywords

Bandwidthbootstrapchange pointconfidence intervalcurve estimationdiscontinuitykernel methodsnonparametric regression

MSC classification

Primary: 62G15: Tolerance and confidence regions
Secondary: 62G09: Resampling methods
Type
Part 2. Estimation methods
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004 , pp. 65 - 79
Copyright
Copyright © Applied Probability Trust 2004 

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References

Cobb, G. W. (1978). The problem of the Nile: conditional solution to a change-point problem. Biometrika 65, 243–251.Google Scholar
Eubank, R. L. and Speckman, P. L. (1993). Confidence bands in nonparametric regression. J. Amer. Statist. Assoc. 88, 1287–1301.Google Scholar
Eubank, R. L. and Speckman, P. L. (1994). Nonparametric estimation of functions with jump discontinuities. In Change-Point Problems , eds. Carlstein, E., Müller, H.-G. and Siegmund, D., Institute of Mathematical Statistics, Hayward, CA.Google Scholar
Eubank, R. L. and Wang, S. (1994). Confidence regions in non-parametric regression. Scand. J. Statist. 21, 147–157.Google Scholar
Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Chapman and Hall, London.Google Scholar
Gijbels, I., Hall, P. and Kneip, A. (1999). On the estimation of jump points in smooth curves. Ann. Inst. Statist. Math. 51, 231–251.CrossRefGoogle Scholar
Hall, P. and Titterington, D. M. (1988). On confidence bands in nonparametric density estimation and regression. J. Multivariate Anal. 27, 228–254.Google Scholar
Härdle, W. and Bowman, A. (1988). Bootstrapping in nonparametric regression: local adaptive smoothing and confidence bands. J. Amer. Statist. Assoc. 83, 102–110.Google Scholar
Härdle, W. and Marron, J. S. (1991). Bootstrap simultaneous error bars for nonparametric regression. Ann. Statist. 19, 778–796.Google Scholar
Hastie, T. J. and Loader, C. (1993). Local regression: automatic kernel carpentry (with discussion). Statist. Sci. 8, 120–143.Google Scholar
Knafl, G., Sacks, J. and Ylvisaker, D. (1985). Confidence bands for regression functions. J. Amer. Statist. Assoc. 80, 683–691.CrossRefGoogle Scholar
Korostelev, A. P. and Tsybakov, A. B. (1993). Minimax Theory of Image Reconstruction (Lecture Notes Statist. 82). Springer, New York.CrossRefGoogle Scholar
Loader, C. (1996). Change point estimation using nonparametric regression. Ann. Statist. 24, 1667–1678.Google Scholar
Mammen, E. (1992). When Does Bootstrap Work? Asymptotic Results and Simulations (Lecture Notes Statist. 77). Springer, New York.Google Scholar
Mcdonald, J. A. and Owen, A. B. (1986). Smoothing with split linear fits. Technometrics 28, 195–208.Google Scholar
Müller, H.-G. (1992). Change-points in nonparametric regression analysis. Ann. Statist. 20, 737–761.CrossRefGoogle Scholar
Müller, H.-G. and Song, K.-S. (1997). Two-stage change-point estimators in smooth regression models. Statist. Prob. Lett. 34, 323–335.Google Scholar
Müller, H.-G. and Stadtmüller, U. (1999). Discontinuous versus smooth regression. Ann. Statist. 27, 299–337.Google Scholar
Raimondo, M. (1996). Modèles en Rupture, Situations non Ergodique et Utilisation de Méthode d'Ondelette. Doctoral Thesis, Université de Paris VII.Google Scholar
Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. John Wiley, New York.Google Scholar
Silverman, B. W. (1985). Some aspects of the spline smoothing approach to nonparametric regression curve fitting (with discussion). J. R. Statist. Soc. B 47, 1–52.Google Scholar
Wahba, G. (1983). Bayesian confidence intervals for the cross-validated smoothing spline. J. R. Statist. Soc. B 45, 133–150.Google Scholar
Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing. Chapman and Hall, London.Google Scholar
Wang, Y. (1995). Jump and sharp cusp detection by wavelets. Biometrika 82, 385–397.Google Scholar