  It is realized that existing powerful tests of goodness-of-fit are all based
on sorted uniforms and, consequently, can suffer from the confounded effect of
different locations and various signal frequencies in the deviations of the
distributions under the alternative hypothesis from those under the null. This
paper proposes circularly symmetric tests that are obtained by circularizing
reweighted Anderson-Darling tests, with the focus on the circularized versions
of Anderson-Darling and Zhang test statistics. Two specific types of
circularization are considered, one is obtained by taking the average of the
corresponding so-called scan test statistics and the other by using the
maximum. To a certain extent, this circularization technique effectively
eliminates the location effect and allows the weights to focus on the various
signal frequencies. A limited but arguably convincing simulation study on
finite-sample performance demonstrates that the circularized Zhang method
outperforms the circularized Anderson-Darling and that the circularized tests
outperform their parent methods. Large-sample theoretical results are also
obtained for the average type of circularization. The results show that both
the circularized Anderson-Darling and circularized Zhang have asymptotic
distributions that are a weighted sum of an infinite number of independent
squared standard normal random variables. In addition, the kernel matrices and
functions are circulant. As a result, asymptotic approximations are
computationally efficient via the fast Fourier transform.
