The Study of Discrete Copula and Quasi-Copular
|School||Anhui University of Engineering|
|Course||Mathematics and Applied Mathematics|
|Keywords||discrete copula idempotent elements permutation discrete quasi-copula GBM(generalized bistochastic matrix) consistency condition complete lattice|
Copula theory was proven to be important in statistical analysis. Quasi-copula, a more general concept, share many properties with copulas, so the study of quasi-copulas has theoretical value. The discrete copulas and discrete quasi-copulas can be regarded as the discreting of copulas and quasi-copulas, so the study of discrete copulas and discrete quasi-copulas can improve the copula theory.First of all, we introduce the basic knowledge of discrete copula and discrete quasi-copula in this paper, and then the discrete copulas and quasi-copulas 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 quasi-copula 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 quasi-copulas. Furthermore, the sequence of discrete quasi-copulas that corresponding to the sequence of GBMs satisfy consistence condition is proved. Because the limit of the sequence of discrete quasi-copulas is a quasi-copula, we get a way to extend the discrete quasi-copula. Thirdly, the multivariate quasi-copulas are considered, and the lattice-theoretic structure of the sets of multivariate quasi-copulas 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.