Reweighted and circularised Anderson-Darling tests of goodness-of-fit

This paper takes a look at omnibus tests of goodness of fit in the context of reweighted Anderson-Darling tests and makes threefold contributions. The first contribution is to provide a geometric understanding. It is argued that the test statistic with minimum variance for exchangeable distributional deviations can serve as a good general-purpose test. The second contribution is to propose better omnibus tests, called circularly symmetric tests and obtained by circularising reweighted Anderson-Darling test statistics or, more generally, test statistics based on the observed order statistics. The resulting tests are called circularised tests. A limited but arguably convincing simulation study on finite-sample performance demonstrates that circularised tests have good performance, as they typically outperform their parent methods in the simulation study. The third contribution is to establish new large-sample results.