The Measures Of Nonlinear Dependence For Random Vectors And Their Application

All things in the world are dependent on each other,which means that interdependence exists universally among the things.It is the task of correlation analysis to explore their dependence.From the perspective of probability and statistics,the joint distribution contains the information of the research object.Copula functions join multivariate distribution to their one-dimensional marginal distribution functions and provide a kind of suitable tools for expressing the characteristics of flexible and complex multivariate joint distributions.The distinctive feature of Copula makes it plays an important role in constructing the measure of dependence and to be widely applied to various fields like economics and finance.By using Copula,this dissertation focuses on the dependence between random variables,the association and tail dependence among random vectors,and constructs effective measures to test,quantify and analyze the corresponding strength of dependence in those association structures.By extending probability density function to Radon-Nikodym derivative w.r.t.a σ-finite product measure,the divergence of essential dependence is proposed based on the generalized Copula density.The divergence of essential dependence is a special case of Kullback-Leibler divergence,reflecting the difference information between joint density and independent density.It is used to figure the essential dependence of inter-group under different grouping for a set of random variables and overcomes the inadequacy of the traditional linear correlation analysis approach to describing the dependence structure.Some examples are given to demonstrate that the divergence of essential dependence can be applied to express the essential dependence under continuous,discrete and mixed distributions and can capture the associations such as MTP2,POD.Spearman’s ρ and Kendall’s τ are classical rank correlation coefficient that measure the dependence between continuous random variables.Copula theory gives them a clear expression as the functionals of Copula.Based on their definitions in continuous case,we extend them to generalized ones.The example of trinomial distribution studies how the generalized ρ and τ vary with the parameter,and points out they are better and more reasonable than Pearson’s correlation coefficient from some views.Based on Spearman’s ρ and Kendall’s τ,two new measures of dependence κ,ι and their asymptotic distributions are established.The simulation results show thatκ which contains the theoretical form and quantitative relationship of the traditional measures of correlation performs better in the statistical inference of independence test.The above measures of dependence are all for two random variables.Furthermore,ρ-measure and τ-measure are provided for measuring and testing interdependence among random vectors.An empirical application to detecting dependence of the daily returns of a portfolio of stocks in NASDAQ illustrates the ρ-measure and τ-measure are the same in judging the positive and negative associations.Combining these two measures with good empirical power in the simulation experiment of independence hypothesis test,it highlights our methods of robustness and effectiveness.Tail dependence is an important consideration in the analysis of financial market risk.To this end,a tail condition that is the random variables are greater than the quantiles under the corresponding given probability levels is introduced.The p-matrix and τ-matrix under that tail condition with their properties and the estimators of their entries(called Spearman’s risk coefficient and Kendall’s risk coefficient,respectively)with the corresponding limiting behaviors of large sample are discussed and figured out.The simulation and application studies show that compared to Spearman’s risk coefficient with large deviation,Kendall’s risk coefficient with robust value and distribution-free assumption is a good approach to quantifying the tail dependence between variables.