  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
circularizing reweighted Anderson-Darling test statistics or, more generally,
test statistics based on the observed order statistics. The resulting tests are
called circularized tests. A limited but arguably convincing simulation study
on finite-sample performance demonstrates that circularized 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.
