TY - JOUR
T1 - Penalized B-spline estimator for regression functions using total variation penalty
AU - Jhong, Jae Hwan
AU - Koo, Ja Yong
AU - Lee, Seong Whan
N1 - Funding 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).
PY - 2017/5/1
Y1 - 2017/5/1
N2 - 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.
AB - 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.
KW - Adaptive estimation
KW - Coordinate descent algorithm
KW - LASSO
KW - Oracle inequalities
KW - Penalized least squares
UR - http://www.scopus.com/inward/record.url?scp=85008598998&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85008598998&partnerID=8YFLogxK
U2 - 10.1016/j.jspi.2016.12.003
DO - 10.1016/j.jspi.2016.12.003
M3 - Article
AN - SCOPUS:85008598998
VL - 184
SP - 77
EP - 93
JO - Journal of Statistical Planning and Inference
JF - Journal of Statistical Planning and Inference
SN - 0378-3758
ER -