Bivariate penalised splines achieve optimal global asymptotic convergence rates similar to tensor product regression splines or thin plate splines, depending on knots and penalty.
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.
Luo Xiao (Wed,) reported a other. Bivariate penalised splines was evaluated on Global asymptotic convergence rate. Bivariate penalised splines achieve optimal global asymptotic convergence rates similar to tensor product regression splines or thin plate splines, depending on knots and penalty.