  Energy distance is a statistical distance between the distributions of random
variables, which characterizes the equality of the distributions. Utilizing the
energy distance, we develop a nonparametric test for the diagonal symmetry,
which is consistent against any fixed alternatives. The test statistic
developed in this paper is based on the difference of two $U$-statistics. By
applying the jackknife empirical likelihood approach, the standard limiting
chi-square distribution with degree freedom of one is established and is used
to determine critical value and $p$-value of the test. Simulation studies show
that our method is competitive in terms of empirical sizes and empirical
powers.
