Estimating Copula and Test of Independence based on a generalized framework of all rank-based Statistics in Bivariate Sample

摘要

Copulas are mathematical objects that fully capture the dependence structureamong random variables and hence, offer a great flexibility in buildingmultivariate stochastic models. In statistics, a copula is used as a generalway of formulating a multivariate distribution in such a way that variousgeneral types of dependence can be represented. In case of bivariate sample,the notion of estimating copula is closely related to that of testingindependence in a bivariate sample, as when the components of the bivariatesample are independent the copula becomes simply product of two uniformdistributions. So apart from non-parametric estimation of copulas we alsoconsidered it relevant to introduce some non-parametric tests to betterunderstand the very essence of copula in the explanation of association betweenthe components. In fact we will develop a general multivariate statistics thatgives rise to a much larger class of non-parametric rank based statistics. Thisclass of statistics can be used in estimation and testing for the associationpresent in the bivariate sample. We choose some representative statistics fromthat class and compared their power in testing independence using simulation asan attempt to choose the best candidate in that class.