The Theory of Copula and the Applications of Statistics of Extremes
Copulas are functions that joint multivariate distribution functions to their one–dimensional marginal distribution functions. The wonderful structure and latent valueof Copulas make their theory develop quickly, and become a popular keyword.By wayof Copula, we can catch the nonlinear and asymmetric dependence between randomvariables and the tail dependence of the distributions, which make Copula give greatplay to analysis of extremes and forecast of extremely phenomenon. This paper mainlytalks about the theory of Copula and the applications of Statistics of Extremes.The upper and lower bounds are the fundamental theory of Copula. When wepossess information about the values of a Copula at points in its domain of definition,the bounds can often be narrowed. We prove the set of Copula, survival Copula, dualof a Copula and the co–Copula together with the composition operation being a Kleinquaternion group, then we give the Fr′echet–Hoe ding bounds and the bounds when wepossess the values of a Copula at some points. We calculate the nonexchangeabilitymeasures of Copula CU and CL, the bounds of Copulas when we possess the valuesof some points, construct four Copulas with maximum nonexchangeability, reveal thestructure feature and property of the four Copulas, and give the calculating formula ofnonexchangeability for the LP distance.FGM Copulas have been widely used in modeling primarily because of their simpleanalytical form. This paper studies the generation and continuation of FGM Copulasbeginning with the methods of constructing Copulas. We construct a new class of generalizedFGM Copulas:C(u; v) = uv+θu~av~b(1-u~m)~c(1-v~n)~d, by proving the necessary andsucient condition when C_θ(u; v) = uv +θf (u)g(v) are Copulas, and study the associationmeasures and dependence properties of these new Copulas,which including manyother types of generalized FGM Copulas appeared in literatures. We make progress ofthe theory of generalized FGM Copulas.Bases on the further study of univariate Extreme and Copula theory, this paperfounds the univariate and bivariate threshold model, studies their applications in busdispatch and the dependence relationships between relief factors of debris flow. Firstly,we found the distribution of bus capacity according to threshold model, and fromthis the optimizing bus scheduling can be given. Secondly, combining the univariatethreshold distribution with Logistic Copula, we construct bivariate threshold model to analyze the extreme dependence between drainage area and drainage height differenceof relief factors of debris flow.Moreover, we study the dependence order of multivariateextreme value distributions using extreme value Copula.