The B-spline copula function is defined by a linear combination of elements of the normalized B-spline basis. We develop a modified EM algorithm, to maximize the penalized log-likelihood function, wherein we use the smoothly clipped absolute deviation (SCAD) penalty function for the penalization term. We conduct simulation studies to demonstrate the stability of the proposed numerical procedure, show that penalization yields estimates with smaller mean-square errors when the true parameter matrix is sparse, and provide methods for determining tuning parameters and for model selection. We analyze as an example a data set consisting of birth and death rates from 237 countries, available at the website, ''Our World in Data,'' and we estimate the marginal density and distribution functions of those rates together with all parameters of our B-spline copula model.

翻译：B样条Copula函数由归一化B样条基元素的线性组合定义。我们开发了一种改进的EM算法，用于最大化惩罚对数似然函数，其中使用平滑剪切绝对偏差（SCAD）惩罚函数作为惩罚项。我们通过模拟研究证明所提数值过程的稳定性，表明当真实参数矩阵稀疏时，惩罚方法能产生均方误差更小的估计，并提供确定调优参数及模型选择的方法。作为实例，我们分析了来自"我们的数据世界"网站中237个国家出生率和死亡率的数据集，估计了这些比率的边际密度函数和分布函数，以及B样条Copula模型的所有参数。