Comparison of Nonparametric Function Estimation Methods for Discontinuous Regression Functions

- Journal title : Korean Journal of Applied Statistics
- Volume 23, Issue 6, 2010, pp.1245-1253
- Publisher : The Korean Statistical Society
- DOI : 10.5351/KJAS.2010.23.6.1245

Title & Authors

Comparison of Nonparametric Function Estimation Methods for Discontinuous Regression Functions

Park, Dong-Ryeon;

Park, Dong-Ryeon;

Abstract

There are two main approaches for estimating the discontinuous regression function nonparametrically. One is the direct approach, the other is the indirect approach. The major goal of the two approaches are different. The direct approach focuses on the overall good estimation of the regression function itself, whereas the indirect approach focuses on the good estimation of jump locations. Apparently, the two approaches are quite different in nature. Gijbels et al. (2007) argue that the comparison of two approaches does not make much sense and that it is even difficult to choose an appropriate criterion for comparisons. However, it is obvious that the indirect approach also has the regression curve estimate as the subsidiary result. Therefore it is necessary to verify the appropriateness of the indirect approach as the estimator of the discontinuous regression function itself. Park (2009a) compared the performance of two approaches through a simulation study. In this paper, we consider a more general case and draw some useful conclusions.

Keywords

Discontinuous regression function;jump detector;jump-preserving smoothing;local M-smoother;

Language

English

References

1.

Bowman, A .W. and Pope, A. (2006). Detecting discontinuities in nonparametric regression curves and surfaces, Statistics & Computing, 16, 377–390.

2.

Burt, D. A. (2000). Bandwidth selection concerns for jump point discontinuity preservation in the regression setting using M-smoothers and the extension to hypothesis testing, Ph. D. dissertation, Virginia Polytechnic Institute and State University, Department of Statistics.

3.

Chu, C. K., Glad, I. K., Godtliebsen, F. and Marron, J. S. (1998). Edge preserving smoothers for image processing (with discussion). Journal of the American Statistical Association, 93, 526–556.

4.

Gijbels, I. and Goderniaux, A. C. (2004). Bandwidth selection for change point estimation in nonparametric regression, Technometics, 46, 76–86

5.

Gijbels, I., Lambert, A. and Qiu, P. (2007). Jump-preserving regression and smoothing using local linear fitting: a compromise, Annals of the Institute of Statistical Mathematics, 59, 235–272.

6.

Park, D. (2008). Estimation of jump points in nonparametric regression, Communications of the Korean Statistical Society, 15, 899–908.

7.

Park, D. (2009a). Comparison of Jump-Preserving Smoothing and Smoothing based on Jump Detector, Communications of the Korean Statistical Society, 16, 519–528.

8.

Park, D. (2009b). Comparison of finite sample properties of jump detectors, Unpublished Manuscript.

9.

Qiu, P. (2005). Image Processing and Jump Regression Analysis, John Wiley & Sons, New Jersey.

10.

Rue, H., Chu, C. K., Godtliebsen, F. and Marron, J. S. (2002). M-smoother with local linear fit, Journal of Nonparametric Statistics, 14, 155–168.

11.

Simpson, D. G., He, X. and Liu, Y. (1998). Comment on Edge-preserving smoothers for image processing by Chu, Glad, Godtliebsen, and Marron, Journal of the American Statistical Association, 93, 544–548.

12.

Wu, J. S. and Chu, C. K. (1993a). Kernel type estimators of jump points and values of a regression function, The Annals of Statistics, 21, 1545–1566.

13.

Wu, J. S. and Chu, C. K. (1993b). Nonparametric function estimation and bandwidth selection for discontinuous regression functions, Statistica Sinica, 3, 557–576.