A copula-based measure for comparing the upper and lower tail probabilities of bivariate distributions
Abstract
Copula is a useful tool for modeling data with complex dependence structure between variables. The aim of this talk is to discuss two topics related to the copulas. First, we present an overview of some basic knowledge about the copulas. Specifically, we provide the definitions of copulas and related measures. Then some basic properties of the copulas are presented. Some well-known examples of the copulas such as Gaussian copula and Archimedean copula are briefly introduced. Second, as work of the speaker, we propose a copula-based measure to compare the upper and lower tail probabilities of bivariate distributions. Some properties of the proposed measure are investigated. It is shown that the limit of the proposed measure as a tuning parameter goes to zero can be expressed in a simple form under certain conditions on copulas. A sample analogue of the proposed measure is given and its weak convergence to a Gaussian process is shown. A nonparametric test based on the sample analogue is presented to test the symmetry of the upper and lower tail probabilities. As an example, the presented measure is applied to stock daily returns of S&P500 and Nikkei225. The work related to the second topic is joint work with Toshinao Yoshiba (Bank of Japan, Japan) and Shinto Eguchi (Institute of Statistical Mathematics, Japan).
