We carry out a study on a penalized regression spline estimator with total variation penalty. In order to provide a spatially adaptive method, we consider total variation penalty for the estimating regression function. This paper adopts B-splines for both numerical implementation and asymptotic analysis because they have small supports, so the information matrices are sparse and banded. Once we express the estimator with a linear combination of B-splines, the coefficients are estimated by minimizing a penalized residual sum of squares. A new coordinate descent algorithm is introduced to handle total variation penalty determined by the B-spline coefficients. For large-sample inference, a nonasymptotic oracle inequality for penalized B-spline estimators is obtained. The oracle inequality is then used to show that the estimator is an optimal adaptive for the estimation of the regression function up to a logarithm factor.
Bibliographical noteFunding Information:
The authors thank the two anonymous reviewers and editors for their valuable comments and suggestions, which led to an improved paper. The research of Ja-Yong Koo was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2013R1A1A2008619). The research of Seong-Whan Lee was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (No. 2012-005741).
© 2017 Elsevier B.V.
- Adaptive estimation
- Coordinate descent algorithm
- Oracle inequalities
- Penalized least squares
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty
- Applied Mathematics