Copula function’s concentration set and its concentrated partition

The research on the local correlation structure of copula function is an attractive topic. This paper investigates bi-variate copula function's local correlation structure by defining its concentration set. The concentration set of a copula function is defined in [0, 1](2) with restrained Lebesgue measure such that the samples of the copula fall in the set with the largest probability. The method for finding the concentration set is provided and the properties of the concentration set are discussed. Based on the concentration set, a concentrated partition of [0, 1](2) for the copula function is introduced, and one measure for quantifying copula function's local correlation is defined by applying our concentrated partition. An empirical study is provided to support our idea of proposing the concentration set.