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Out of the many different types of symmetries of a continuous bivariate joint probability density function, focus here is on the central symmetry and symmetry about an axis. Tests for both types of symmetries are proposed and evaluated in the literature. However interestingly, investigation into tests for and a general discussion of either central symmetry or symmetry about an axis don’t seem to observe the relationships between these two symmetries. Here we first show that testing for central symmetry is mathematically equivalent to testing for symmetry about an axis and vice versa. This equivalence is then exhibited through simulation by comparing a recently proposed test for testing symmetry about an axis in Rao and Raghunath with a recently proposed test for testing central symmetry in Einmahl and Gan and vice versa. It is observed that the test of symmetry about an axis say, the  X -axis, considered in Rao and Raghunath actually tests that the median regression of  Y  on  X  is theaxis and hence the test has no power against the alternative when the bivariate probability density function is not symmetric about theaxis but the median regression of  Y  on  X  is theaxis. Therefore a new test, resulting out of a modification of the above mentioned test of symmetry about an axis, is proposed which does detect asymmetry about an axis even if that axis happens to be the median regression line. Simulation study exhibits the better performance of the new test compared to the test for testing symmetry about an axis in Rao and Raghunath; simulation also show that, if used for testing the central symmetry, it as good as the test of central symmetry in Einmahl and Gan. At the end, the new test is applied to a real medical data set.

Testing for central symmetry and symmetry about an axis