Abstract

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 the $X-$axis and hence the test has no power against the alternative when the bivariate probability density function is not symmetric about the $X-$axis but the median regression of Y on X is the $X-$axis. 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.

摘要

在连续双变量联合概率密度函数的许多不同类型的对称性中，这里的重点是中心对称性和关于轴的对称性。文献中提出并评估了两种类型的对称性测试。然而有趣的是，对中心对称或轴对称的测试和一般讨论的调查似乎并没有观察到这两个对称之间的关系。在这里，我们首先表明测试中心对称性在数学上等同于测试关于轴的对称性，反之亦然。然后，通过比较最近提出的用于测试 Rao 和 Raghunath 中的轴对称性的测试与最近提出的用于测试 Einmahl 和 Gan 中的中心对称性的测试，反之亦然，通过模拟展示了这种等价性。Rao 和 Raghunath 考虑的X轴实际上测试了Y在X上的中值回归是$X-$轴，因此当双变量概率密度函数关于$X-$轴，但Y在X上的中值回归是$X-$轴。因此，提出了一种新的测试，该测试是对上述关于轴的对称性测试的修改而产生的，它确实检测关于轴的不对称性，即使该轴恰好是中值回归线。仿真研究表明，与 Rao 和 Raghunath 中关于轴对称性的测试相比，新测试的性能更好；仿真还表明，如果用于检验中心对称性，它与Einmahl和Gan中的中心对称性检验一样好。最后，将新测试应用于真实的医学数据集。