Local linear fitting and improved estimation near peaks. (English) Zbl 1177.62048

Summary: The authors look into the problem of estimating regression functions that exhibit jump irregularities in the first derivative. They investigate the behaviour of the bias in the local linear fit and show the superior performance of appropriate one-sided versions of the local linear fit near such irregularities. They then propose an improved estimation procedure based on data-driven selection of a conventional or one-sided local linear fit according to a residual sum of squares type of criterion. The authors provide theoretical results and illustrate the method both on simulated and real-life data examples.


62G08 Nonparametric regression and quantile regression
65C60 Computational problems in statistics (MSC2010)


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[1] Bowman, Detecting discontinuities in nonparametric regression curves and surfaces, Statistics and Computing 16 pp 377– (2006)
[2] Cheng, Kernel methods for optimal change-points estimation in derivatives, Journal of Computational and Graphical Statistics 17 pp 1– (2008)
[3] Cline, Nonparametric estimation of regression curves with discontinuous derivatives, Journal of Statistical Research 29 pp 17– (1995)
[4] L. Desmet & I. Gijbels (2009). Local linear fitting and improved estimation near peaks. Technical Report. Section of Statistics, Department of Mathematics, Katholieke Universiteit Leuven. Available at http://wis.kuleuven.be/stat/publications.html · Zbl 1177.62048
[5] Donoho, Ideal spatial adaptation by wavelet shrinkage, Biometrika 81 pp 425– (1994) · Zbl 0815.62019
[6] Donoho, Adapting to Unknown Smoothness via Wavelet Shrinkage, Journal of the American Statistical Association 90 pp 1200– (1995) · Zbl 0869.62024
[7] Fan, Local Polynomial Modelling and Its Applications (1996) · Zbl 0873.62037
[8] Gijbels, Data-driven discontinuity detection in derivatives of a regression function, Communications in Statistics-Theory and Methods 33 pp 851– (2004) · Zbl 1218.62031
[9] Gijbels, Bootstrap test for change points in nonparametric regression, Journal of Nonparametric Statistics 16 pp 591– (2004) · Zbl 1148.62304
[10] Gijbels, Jump-preserving regression and smoothing using local linear fitting: A compromise, The Annals of the Institute of Statistical Mathematics 59 pp 235– (2007)
[11] Hall, Edge-preserving and Peak-preserving smoothing, Technometrics 34 pp 429– (1992)
[12] E. Hermann (2003). Lokern: An R package for kernel smoothing. Available online at http://www.r-project.org/.
[13] Kovac, Smooth functions and local extreme values, Computational Statistics and Data Analysis 51 pp 5156– (2007) · Zbl 1162.62358
[14] A. Lambert (2005). Nonparametric estimations of discontinuous curves and surfaces. PhD thesis, Institut de Statistique, Université Catholique de Louvain, Belgium.
[15] McDonald, Smoothing with split linear fits, Technometrics 28 pp 195– (1986) · Zbl 0626.65010
[16] Müller, Adaptive nonparametric peak estimation, The Annals of Statistics 17 pp 1053– (1989)
[17] Müller, Change-points in nonparametric regression analysis, The Annals of Statistics 20 pp 737– (1992)
[18] Qiu, A jump-preserving curve fitting procedure based on local piecewise-linear kernel estimation, Journal of Nonparametric Statistics 15 pp 437– (2003) · Zbl 1054.62047
[19] Qiu, A local polynomial jump detection algorithm in nonparametric regression, Technometrics 40 pp 141– (1998)
[20] Russell, Nursing behaviors of beluga calves born in captivity, Zoo Biology 16 pp 247– (1997)
[21] Simonoff, Smoothing Methods in Statistics (1996)
[22] Wu, Kernel type estimation of jump points and values of regression function, The Annals of Statistics 21 pp 1545– (1993)
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