The Study of Discrete Copula and QuasiCopular 

Author  NingLong 
Tutor  WangChuanYu 
School  Anhui University of Engineering 
Course  Mathematics and Applied Mathematics 
Keywords  discrete copula idempotent elements permutation discrete quasicopula GBM(generalized bistochastic matrix) consistency condition complete lattice 
CLC  O211.6 
Type  Master's thesis 
Year  2011 
Downloads  10 
Quotes  0 
Copula theory was proven to be important in statistical analysis. Quasicopula, a more general concept, share many properties with copulas, so the study of quasicopulas has theoretical value. The discrete copulas and discrete quasicopulas can be regarded as the discreting of copulas and quasicopulas, so the study of discrete copulas and discrete quasicopulas can improve the copula theory.First of all, we introduce the basic knowledge of discrete copula and discrete quasicopula in this paper, and then the discrete copulas and quasicopulas are studied further by using the theory of combinatorial mathematics, matrix and lattice. The specific content include three aspects as the following:Firstly, the numbers of irreducible discrete copulas are studied from the standpoint of its idempotent elements by using combinatorial mathematics. In addition, because the permutation matrix is a special Boolean matrix, the question that the result of three basic Boolean operations between permutation matrices is still or not a permutation matrix is also discussed. Secondly, according to the relationship between GBM and discrete copula, the extension of discrete quasicopula is studied from the standpoint of matrix. Specifically, for a given GBM, we get a way to construct a sequence of GBMs, and then correspondingly the sequence of discrete quasicopulas. Furthermore, the sequence of discrete quasicopulas that corresponding to the sequence of GBMs satisfy consistence condition is proved. Because the limit of the sequence of discrete quasicopulas is a quasicopula, we get a way to extend the discrete quasicopula. Thirdly, the multivariate quasicopulas are considered, and the latticetheoretic structure of the sets of multivariate quasicopulas is studied especially by applying the theory of lattice.At last to summary the content of this paper mentioned before and point out the direction of research in the next step.