Estimation of 2D jump location curve and 3D jump location surface in nonparametric regression

Authors:
Chih-Kang Chu;Jhao-Siang Siao;Lih-Chung Wang;Wen-Shuenn Deng
Affiliations:
Department of Applied Mathematics, National Dong Hwa University, Hualien, Taiwan 974;Department of Applied Mathematics, National Dong Hwa University, Hualien, Taiwan 974;Department of Applied Mathematics, National Dong Hwa University, Hualien, Taiwan 974;Department of Statistics, Tamkang University, Tamsui, Taiwan 251
Venue:
Statistics and Computing
Year:
2012

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Abstract

A new procedure is proposed to estimate the jump location curve and surface in the two-dimensional (2D) and three-dimensional (3D) nonparametric jump regression models, respectively. In each of the 2D and 3D cases, our estimation procedure is motivated by the fact that, under some regularity conditions, the ridge location of the rotational difference kernel estimate (RDKE; Qiu in Sankhy驴 Ser. A 59, 268---294, 1997, and J. Comput. Graph. Stat. 11, 799---822, 2002; Garlipp and Müller in Sankhy驴 Ser. A 69, 55---86, 2007) obtained from the noisy image is asymptotically close to the jump location of the true image. Accordingly, a computational procedure based on the kernel smoothing method is designed to find the ridge location of RDKE, and the result is taken as the jump location estimate. The sequence relationship among the points comprising our jump location estimate is obtained. Our jump location estimate is produced without the knowledge of the range or shape of jump region. Simulation results demonstrate that the proposed estimation procedure can detect the jump location very well, and thus it is a useful alternative for estimating the jump location in each of the 2D and 3D cases.