Pairwise distance-based tests for conditional symmetry

In this paper, we develop a pairwise distance-based testing procedure for conditional symmetry of a random vector given another random vector and apply it to the cases with given and unknown center that is a parametric regression function, respectively. The resulting tests are moments-based and thus the curse of dimensionality can then be greatly alleviated. The asymptotic properties of the test statistics are investigated. The tests can detect local alternatives distinct from the null at a fastest possible convergence rate in hypothesis testing. To determine critical values, a Monte Carlo-based approximation to the limiting null distributions is suggested. We prove that the approximation works even under local alternative hypotheses. Some simulation studies and a real data example are conducted to examine the performance of the tests.