Dynamic Tail Dependence And The Construction Of A Kind Of Multivariate Extreme Value Distribution

It is well-known that the asymptotic distribution of the multivariate extremes,i.e.,the multivariate extreme value distribution,is determined by the asymptotic tail behavior of the underlying distributions,and so is the probability that extreme events occur simultaneously.Measuring the tail dependence of a multivariate distribution is the key to quantitative risk assessment,it hence finds important applications in fields such as finance and climate science.Mastering the tail dependence property of specific distribution families is beneficial to the accurate selection of models in data analysis,and constructing live and tractable tail dependence models is the cornerstone for the accurate measurement of dependence.The tail dependence function is an important measure in the study of tail dependence,while multivariate extreme value distributions are a common choice for modeling the tail dependence and multivariate extremes of multivariate distributions.In this thesis,we propose a new concept termed the dynamic tail dependence function and investigate the asymptotic tail behavior of the bivariate Gaussian distributions and skew-normal distributions with correlation coefficients satisfying the Hiisler-Reiss condition in terms of this new dependence function;we also construct a kind of tree-structured multivariate extreme value distributions based on Markov trees,which can be used in the modeling of tail dependence of multi-dimensional or high dimensional data.Details are as follows:Chapter two considers the dynamic tail dependence functions of bivariate Gaussian distributions with correlation coefficients satisfying the Hiisler-Reiss condition.It is known that the tail dependence coefficient of a bivariate Gaussian random vector with a constant correlation coefficient is zero and the rate of decay of the tail dependence is linked with a regularly varying function.By introducing dynamic correlation coefficients to the bivariate Gaussian distribution,the partial maxima of the resulting bivariate Hiisler-Reiss model could possess asymptotic dependent limit distributions.Under the setting that the bivariate Gaussian random vectors have dynamic correlation coefficient satisfying the Hiisler-Reiss condition,we derived its dynamic tail dependence function and establish the convergence rates to the dynamic tail dependence functions under some refined HiislerReiss conditions.By-products are the tail orders and tail order functions with related expansions.Chapter three focuses on the tail behavior of bivariate skew-normal distribution with a dynamic correlation coefficient.For a bivariate skew-normal random vector with equal skewness and dynamic correlation coefficient satisfying the Hiisler-Reiss condition,the explicit expression of the dynamic tail dependence coefficient and the convergence rates to the dynamic tail dependence coefficient are obtained.By introducing a dynamic correlation coefficient,this bivariate skew-normal distribution family possesses a richer dependence structure than the skew-normal distributions with constant correlation coefficients:the tail may be asymptotic independent,asymptotic dependent or asymptotic completely dependent,depending on the Hiisler-Reiss condition.Numerical analysis is performed to illustrate our findings.Chapter four discusses the construction and statistical inference of a kind of treestructured multivariate extreme value distribution.Multivariate extreme value distributions are a common choice for modeling multivariate extremes.However,the construction of flexible and parsimonious models is still challenging.For a multivariate extreme value distribution G,we propose to combine its bivariate marginal extreme value distributions into a Markov random field Y with respect to a tree.Although in general not an extreme value distribution itself,this Markov tree Y is attracted by a multivariate extreme value distribution GM.The distribution GM serves as a tree-based approximation to the extreme value distribution G.Given a set of data with distribution F belonging to the domain of attraction of G,we learn an appropriate tree structure by Prim’s algorithm with estimated pairwise upper tail dependence coefficients or Kendall’s tau values as edge weights.The distributions of pairs of connected variables on the built Markov tree can be fitted in various ways and finally we obtain an estimate of the resulting tree-structured extreme value distribution.We prove in this thesis that:if G itself has a tree-based dependence structure,with probability tending to 1,the set of maximum dependence trees weighted by the upper tail dependence coefficients contains the true tree structure;and the joint estimator of the parameters of all the edge-wise bivariate dependence structures is still asymptotically normal,provided all the edge-wise estimators have a certain asymptotic expansion.The method is illustrated by a simulation study and an application on river discharge data from the upper Danube basin to estimate the flooding probability.