The Algorithms Of Regular Vine Copulas Modeling,Sampling And Testing With Application In Sampling Of Multivariate Count Variables

Regular vine copula provides rich models for dependence structure modeling.It combines vine structures and families of bivariate copulas to construct a number of mul-tivariate distributions that can model a wide range dependence patterns with different tail dependence for different pairs.Two special cases of regular vine copulas,C-vine and D-vine copulas,have been extensively investigated in the literature.In this thesis,we develop the algorithms of modeling,sampling and testing for a more generalized regular vine copula(R-vine for short).R-vine modeling algorithm searches for the R-vine structure which maximizes the vine tree dependence in a sequen-tial way.The maximum likelihood estimation algorithm takes the sequential estimations as initial values and uses L-BFGS-B algorithm for the likelihood value optimization.R-vine sampling algorithm traverses all edges of the vine structure from the last tree in a recursive way,and generates the marginal samples on each edge according to some nested conditions.Goodness-of-fit testing algorithm first generates Rosenblatt's trans-formed data E,then tests the hypothesis H0*:E?C? by using Anderson-Darling statistic,where C? is the independence copula.Bootstrap method is used to compute an adjusted p-value of the empirical distribution of replications of Anderson-Darling statistic.The computing of related functions of copulas such as cumulative distribution functions,H-functions and inverse H-functions often meets with the problem of over-flow.We solve this problem by reinvestigating the following six families of bivariate copulas:Normal,Student t,Clayton,Gumbel,Frank and Joe's copulas.Approxima-tions of the above related functions of copulas are given when the overflow occurs in the computation.The problem of sampling multivariate count variariables has practical significance.Erhardt raised an algorithm for sampling multivariate count random variables based on C-vine copulas,and the parameter pi,j|D of edge ei,j|D of the C-vine structure are estimated by optimizing the difference between sample partial correlation ?ij|D and the partial correlation ?ij|D calculated from correlation matrix by the Pearson recurrence formula.We raise the concept of marginal regular vine copulas,which leads to directly optimize difference of sample correlation ?ij and the targeted correlation ?ij for pairs of variables.Three simulation studies are carried to give comparison of the new sampling algorithm to Erhardt's and the Naive method.Three Python packages are implemented.The pyvine package implements the algorithms for regular vine copulas modeling,sampling and testing.The bvcopula package implements high performance and precision of routines including c.d.f,p.d.f,sampling,H-function and inverse H-function of six common seen bivariate copulas families:Normal,Student t,Clayton,Gumbel,Frank and Joe's copulas.The countvar package implemented the sampling algorithm from multivariate count variables with specified marginal distributions and correlation matrix and routines of four common seen univariate discrete variables.