  We propose a copula-based measure of asymmetry between the lower and upper
tail probabilities of bivariate distributions. The proposed measure has a
simple form and possesses some desirable properties as a measure of asymmetry.
The limit of the proposed measure as the index goes to the boundary of its
domain can be expressed in a simple form under certain conditions on copulas. A
sample analogue of the proposed measure for a sample from a copula is presented
and its weak convergence to a Gaussian process is shown. Another sample
analogue of the presented measure, which is based on a sample from a
distribution on $\mathbb{R}^2$, is given. Simple methods for interval
estimation and nonparametric testing based on the two sample analogues are
presented. As an example, the presented measure is applied to daily returns of
S&P500 and Nikkei225.
