that all existing strong tests of goodness-of-fit are based on sorted uniforms and, as a result, can suffer from confounding effects of different locations and different signal frequencies in the deviation of the distribution under the alternative hypothesis. is recognized as null. In this paper, we focus on the circular version of the Anderson-Darling and Chang test statistic and propose a circular symmetric test obtained by circularizing the reweighted Anderson-Darling test. Two specific types of circularization are considered. One is obtained by averaging the corresponding so-called scan test statistics, the other by using the maximum value. To some extent, this circularization technique can effectively eliminate position effects and concentrate the weights on different signal frequencies. Limited but arguably compelling simulation studies on finite-sample performance show that the cycled Zhang method outperforms the cycled Anderson-Darling method, and that the cycled test It shows that it is better than the parent’s method. Large-sample theoretical results are also obtained for average-type circularizations. The results show that both the cycled Anderson-Darling and the cycled Zhang have asymptotic distributions that are weighted sums of squares of an infinite number of independent standard normal random variables. Additionally, kernel matrices and functions are cyclic. As a result, the asymptotic approximation is computationally efficient via the fast Fourier transform.

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