ABSTRACT Sample entropy based tests, methods of sieves and Grenander estimation type procedures are known to be very efficient tools for assessing normality of underlying data distributions, in one-dimensional nonparametric settings. Recently, it has been shown that the density based empirical likelihood (EL) concept extends and standardizes these methods, presenting a powerful approach for approximating optimal parametric likelihood ratio test statistics, in a distribution-free manner. In this paper, we discuss difficulties related to constructing density based EL ratio techniques for testing bivariate normality and propose a solution regarding this problem. Toward this end, a novel bivariate sample entropy expression is derived and shown to satisfy the known concept related to bivariate histogram density estimations. Monte Carlo results show that the new density based EL ratio tests for bivariate normality behave very well for finite sample sizes. To exemplify the excellent applicability of the proposed approach, we demonstrate a real data example.

中文翻译：

---

一种用于检验双变量正态性的基于密度的经验似然方法

摘要 众所周知，基于样本熵的测试、筛法和格伦南德估计类型程序是在一维非参数设置中评估基础数据分布的正态性的非常有效的工具。最近，已经表明基于密度的经验似然 (EL) 概念扩展并标准化了这些方法，提供了一种以无分布的方式逼近最佳参数似然比检验统计量的强大方法。在本文中，我们讨论了与构建用于测试双变量正态性的基于密度的 EL 比率技术相关的困难，并针对该问题提出了解决方案。为此，导出并显示了一种新的双变量样本熵表达式，以满足与双变量直方图密度估计相关的已知概念。Monte Carlo 结果表明，新的基于密度的双变量正态性 EL 比率测试对于有限的样本大小表现得非常好。为了举例说明所提出方法的出色适用性，我们展示了一个真实的数据示例。