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 pseudo-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样条基函数的线性组合定义。我们开发了一种改进的EM算法，用于最大化惩罚伪似然函数，其中惩罚项采用光滑削边绝对偏离（SCAD）惩罚函数。我们通过模拟研究验证了所提数值过程的稳定性，证明了当真实参数矩阵稀疏时，惩罚化能够产生具有更小均方误差的估计量，并提供了确定调优参数和进行模型选择的方法。作为示例，我们分析了来自“Our World in Data”网站的237个国家的出生率与死亡率数据集，估计了这些比率的边缘密度函数与分布函数，以及我们B样条Copula模型的所有参数。