In the field of finance, insurance, and system reliability, etc., it is often of interest to measure the dependence among variables by modeling a multivariate distribution using a copula. The copula models with parametric assumptions are easy to estimate but can be highly biased when such assumptions are false, while the empirical copulas are non-smooth and often not genuine copula making the inference about dependence challenging in practice. As a compromise, the empirical Bernstein copula provides a smooth estimator but the estimation of tuning parameters remains elusive. In this paper, by using the so-called empirical checkerboard copula we build a hierarchical empirical Bayes model that enables the estimation of a smooth copula function for arbitrary dimensions. The proposed estimator based on the multivariate Bernstein polynomials is itself a genuine copula and the selection of its dimension-varying degrees is data-dependent. We also show that the proposed copula estimator provides a more accurate estimate of several multivariate dependence measures which can be obtained in closed form. We investigate the asymptotic and finite-sample performance of the proposed estimator and compare it with some nonparametric estimators through simulation studies. An application to portfolio risk management is presented along with a quantification of estimation uncertainty.
翻译:在金融、保险和系统可靠性等领域,通常有兴趣通过使用千叶草板模拟多变量分布模型来测量变量之间的依赖性。带有参数假设的千叶形模型很容易估计,但如果这些假设是假的,则可能高度偏差,而经验性的千叶形模型本身不光滑,而且往往不是真正的千叶色,从而得出在实践中具有挑战性的依赖性的推论。作为一种折中,经验性的伯恩斯坦千叶色板提供了一个平滑的估测符,但调控参数的估算仍然难以实现。在本文中,我们通过使用所谓的实证式棋盘棋盘焦云构建了一个等级的经验性海湾模型,使得能够对平滑的千叶色功能进行任意的估算。基于多叶色伯恩斯坦多面形图谱系的拟议的估计符本身是一个真正的千叶色相,其尺寸变化度的选择取决于数据。我们还表明,拟议的千叶色色图测量仪提供了以封闭形式获得的若干多变量依赖性措施的更准确的估计。我们用某种等级的实验性与定数和定数分析模型的估算结果来比较。