基于copula的相關性測度

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
E FDM of variables in crime dataset