Symmetry test

A statistical test for testing the hypothesis $H_0$ that a one-dimensional probability density is symmetric about zero.

Let the hypothesis of symmetry $H_0$ be that the probability density $p(x)$ of the probability law of independent random variables $X_1,\ldots,X_n$ is symmetric about zero, that is, $p(x)=p(-x)$ for any $x$ from the domain of definition of $p(x)$. Any statistical test intended for testing $H_0$ is called a symmetry test.

Most often the hypothesis $H_1$ that all the random variables $X_1,\ldots,X_n$ have probability density $p(x-\Delta)$, $\Delta\neq0$, is considered as the alternative to $H_0$. In other words, according to $H_1$ the probability density of $X_i$ is obtained by shifting the density $p(x)$ along the $x$-axis by a distance $|\Delta|$, to the right or left according to the sign of $\Delta$. If the sign of the displacement $\Delta$ is known, then $H_1$ is called one-sided, otherwise it is called two-sided. A simple example of a symmetry test is given by the sign test.

Usually a randomization test is used for testing symmetry.

References

[1]	Z. Sidak, "Theory of rank tests" , Acad. Press (1967)
[2]	M.G. Kendall, A. Stuart, "The advanced theory of statistics" , 2. Inference and relationship , Griffin (1979)