基于Copula的相关性测度/江西财经大学东亿学术论丛

Copula在应用统计领域 ，如金融、气象、水文等有 广泛的应用。本书从Copula 视角介绍了变量间几种相关 性的度量，着重讨论了变量 之间函数型关系强弱的基于 Copula的度量。 变量间的函数型关系是 一种较为广泛的概念，既包 括了常见的线性关系、非线 性单调关系，又包括了目前 较少讨论的非单调关系。因 此本书的工作具有广泛的适 用性，同时也为非线性关系 的度量提供了另一种思路。 函数型关系是一个比线性关 系、单调型关系更广泛的概 念，本书分别针对离散型和 连续型函数关系作了讨论： 对离散型变量，构造了几种 基于Subcopula的测度，并 讨论了这些测度的理论性质 ；对连续型变量的测度，主 要从非参数核密度估计入手 ，构造了其非参数估计，讨 论了其渐进性质，并给出了 数值模拟结果。

1 Outline and Summary 1.1 Introduction 1.2 Outline 2 Statistical Modeling and Measurement of Association 2.1 The concept of copulas 2.2 Nonparametric estimations of copula 2.2.1 An overview of empirical processes 2.2.2 Nonparametric estimation via the empirical copula 2.2.3 Functional delta-method and hadamard differentiability 2.2.4 Weak convergence of the empirical copula process 2.2.5 Nonparametric kernel estimations 2.2.6 Bias and variance of kernel density estimator 2.2.7 Optimal bandwith 2.3 Measures of association and dependence 2.3.1 Pearson's corelation coefficient 2.3.2 Spearman's ρ and Kendall's τ 2.3.3 The measure for mutual complete dependence 2.3.4 The * operator and the measure of mutual complete dependence 3 A Measure for Positive Quadrant Dependence 4 Measures for Discrete MCD and Functional Dependence 4.1 The measure of MCD through conditional distributions 4.2 The measure of MCD through a subcopula 4.3 Comparison to Siburg and Stoimenov's measure of MCD 4.3.1 Extension using E-process 4.3.2 Bilinear extension 4.4 Remarks on measures of dependence 4.5 Other measures 4.5.1 The measure μ20 4.5.2 The measure λ 4.5.3 Construction of the measure 4.5.4 Proofs of the construction of λ 5 Nonparametric Estimation of the Measure of Functional Dependence 5.1 Nonparametric estimation through the density of copula 5.1.1 Estimating with pseudo-observations 5.1.2 Kernel estimation through copula density functions 5.1.3 Asymptotic behavior of the estimator of functional dependence 5.2 Nonparametric estimation of the measure of MCD via copula 5.3 Simulation results 6 Implementation and Simulations 6.1 Choosing the evaluation grid 6.2 Simulation 6.3 Comparison of measures 7 Application 8 Discussion References Appendix A List of Symbols B Calculation of the Measure of PQD C Beta Kernel Estimation D Kernel Estimation