• The Korean Journal of Applied Statistics
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• v.23 no.6
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• pp.1245-1253
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• 2010

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.

• Communications for Statistical Applications and Methods
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• v.16 no.3
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• pp.519-528
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• 2009

This paper deals with nonparametric estimation of discontinuous regression curve. Quite number of researches about this topic have been done. These researches are classified into two categories, the indirect approach and direct approach. The major goal of the indirect approach is to obtain good estimates of jump locations, whereas the major goal of the direct approach is to obtain overall good estimate of the regression curve. Thus it seems that two approaches are quite different in nature, so people say that the comparison of two approaches does not make much sense. Therefore, a thorough comparison of them is lacking. However, even though the main issue of the indirect approach is the estimation of jump locations, it is too obvious that we have an estimate of regression curve as the subsidiary result. The point is whether the subsidiary result of the indirect approach is as good as the main result of the direct approach. The performance of two approaches is compared through a simulation study and it turns out that the indirect approach is a very competitive tool for estimating discontinuous regression curve itself.

• Proceedings of the Korean Statistical Society Conference
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• 2002.11a
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• pp.103-108
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• 2002

We consider an estimation of discontinuous variance function in nonparametric heteroscedastic random design regression model. We first propose estimators of a change point and jump size in variance function and then construct an estimator of entire variance function. We examine the rates of convergence of these estimators and give results on their asymptotics. Numerical work reveals that the effectiveness of change point analysis in variance function estimation is quite significant.

• Journal of the Korean Data and Information Science Society
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• v.27 no.1
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• pp.111-120
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• 2016

In usual, the discontinuous variance function was estimated nonparametrically using a kernel type estimator with data sets split by an estimated location of the change point. Kang et al. (2000) proposed the Gasser-$M{\ddot{u}}ller$ type kernel estimator of the discontinuous regression function using the adjusted observations of response variable by the estimated jump size of the change point in $M{\ddot{u}}ller$ (1992). The adjusted observations might be a random sample coming from a continuous regression function. In this paper, we estimate the variance function using the Nadaraya-Watson kernel type estimator using the adjusted squared residuals by the estimated location of the change point in the discontinuous variance function like Kang et al. (2000) did. The rate of convergence of integrated squared error of the proposed variance estimator is derived and numerical work demonstrates the improved performance of the method over the exist one with simulated examples.

• Communications for Statistical Applications and Methods
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• v.15 no.6
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• pp.899-908
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• 2008

If the regression function has jump points, nonparametric estimation method based on local smoothing is not statistically consistent. Therefore, when we estimate regression function, it is quite important to know whether it is reasonable to assume that regression function is continuous. If the regression function appears to have jump points, then we should estimate first the location of jump points. In this paper, we propose a procedure which can do both the testing hypothesis of discontinuity of regression function and the estimation of the number and the location of jump points simultaneously. The performance of the proposed method is evaluated through a simulation study. We also apply the procedure to real data sets as examples.

• Journal of the Korean Statistical Society
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• v.35 no.1
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• pp.1-23
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• 2006

In this paper we consider an estimation of the discontinuous variance function in nonparametric heteroscedastic random design regression model. We first propose estimators of the change point in the variance function and then construct an estimator of the entire variance function. We examine the rates of convergence of these estimators and give results for their asymptotics. Numerical work reveals that using the proposed change point analysis in the variance function estimation is quite effective.

• Communications for Statistical Applications and Methods
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• v.19 no.4
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• pp.579-590
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• 2012

Local linear jump detection in a discontinuous regression function involves the choice of the bandwidth and the performance of a local linear jump detector depends heavily on the choice of the bandwidth. However, little attention has been paid to this important issue. In this paper we propose two fully data adaptive bandwidth selection methods for a local linear jump detector. The performance of the proposed methods are investigated through a simulation study.

• Communications for Statistical Applications and Methods
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• v.16 no.6
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• pp.1047-1054
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• 2009

Local smoothing jump detection procedure is a popular method for detecting jump locations and the performance of the jump detector heavily depends on the choice of the bandwidth. However, little work has been done on this issue. In this paper, we propose the bootstrap bandwidth selection method which can be used for any kernel-based or local polynomial-based jump detector. The proposed bandwidth selection method is fully data-adaptive and its performance is evaluated through a simulation study and a real data example.

• Journal of the Korean Data and Information Science Society
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• v.25 no.1
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• pp.87-95
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• 2014

In the regression model, Kang and Huh (2006) studied the estimation of the discontinuous variance function using the Nadaraya-Watson estimator with the squared residuals. The local linear estimator of the log-variance function, which may have the whole real number, was proposed by Huh (2013) based on the kernel weighted local-likelihood of the ${\chi}^2$-distribution. Chen et al. (2009) estimated the continuous variance function using the local linear fit with the log-squared residuals. In this paper, the estimator of the discontinuous log-variance function itself or its derivative using Chen et al. (2009)'s estimator. Numerical works investigate the performances of the estimators with simulated examples.

• Journal of the Korean Data and Information Science Society
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• v.17 no.3
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• pp.707-716
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• 2006

When the regression function is discontinuous at a point, the variance function is usually discontinuous at the point. In this case, we had better propose a test for the existence of a discontinuity point with the regression function rather than the variance function. In this paper we consider that the variance function only has a discontinuity point. We propose a nonparametric test for the existence of a discontinuity point with the second moment function since the variance function and the second moment function have the same location and jump size of the discontinuity point. The proposed method is based on the asymptotic distribution of the estimated jump size.