Powerful Omnibus Tests of Goodness-of-Fit

This paper takes a look at omnibus tests of goodness-of-test in the context of reweighted Anderson-Darling tests and makes three fold contributions. The first contribution is to provide a geometric understanding. It is argued that the test statistic with minimum variance can serve as a good general-purpose test. The second contribution is to propose better omnibus tests, called circularly symmetric tests and obtained by circularizing reweighted Anderson-Darling tests. A limited but arguably convincing simulation study on finite-sample performance demonstrates that the circularized tests outperform their parent methods. The third contribution is to establish new large-sample results. It is shown that like Anderson-Darling, the minimum-variance test statistic and two circularly symmetric test statistics under the null has the same distribution as that of a weighted sum of an infinite number of independent squared normal random variables.