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.

中文翻译：

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二元惩罚样条的渐近线

摘要 我们研究了使用张量积样条和平滑度惩罚的二元惩罚样条类。类似于 Claeskens, G.、Krivobokova, T. 和 Opsomer, JD [(2009), 'Asymptotic Properties of Penalized Spline Estimators', Biometrika, 96(3), 529–544] 对于单变量惩罚样条，我们表明，取决于结点和惩罚的数量，双变量惩罚样条的全局渐近收敛率要么类似于张量积回归样条，要么类似于薄板样条。在每种情况下，双变量惩罚样条都被发现在 Stone 意义上的速率最优 [(12, 1982), 'Optimal Global Rates of Convergence for Nonparametric Regression', The Annals of Statistics, 10(4), 1040–1053 ] 对于具有适当平滑度的相应类函数。对于使用少量节点的场景，我们获得局部渐近偏差和方差的表达式，并导出逐点和均匀渐近正态性。理论结果适用于张量积回归样条。