Asymptotics of bivariate penalised splines

ABSTRACT

We study the class of bivariate penalised splines that use tensor product splines and a smoothness penalty. Similar to Claeskens, G., Krivobokova, T., and Opsomer, J.D. [(2009), ‘Asymptotic Properties of Penalised Spline Estimators’, Biometrika, 96(3), 529–544] for the univariate penalised splines, we show that, depending on the number of knots and penalty, the global asymptotic convergence rate of bivariate penalised splines is either similar to that of tensor product regression splines or to that of thin plate splines. In each scenario, the bivariate penalised splines are found rate optimal in the sense of Stone, C.J. [(12, 1982), ‘Optimal Global Rates of Convergence for Nonparametric Regression’, The Annals of Statistics, 10(4), 1040–1053] for a corresponding class of functions with appropriate smoothness. For the scenario where a small number of knots is used, we obtain expressions for the local asymptotic bias and variance and derive the point-wise and uniform asymptotic normality. The theoretical results are applicable to tensor product regression splines.

KEYWORDS:

• B-splines
• P-splines
• regression spline
• smoothing
• tensor product splines

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 62G05
• 62G08
• 62G15
• 62G20

Acknowledgments

The author wishes to thank the anonymous referees and the associate editor for their constructive comments, which led to significant improvement of the paper. The author would like to thank the Editor-in-Chief Professor Jun Shao for help and encouragement.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Luo Xiao  http://orcid.org/0000-0001-8707-0914