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.

翻译：现有强效拟合优度检验均基于排序均匀分布，因而可能受到不同位置和信号频率在备择假设与零假设分布偏差中的混杂效应影响。本文提出通过循环化加权安德森-达林检验得到的循环对称检验，重点研究安德森-达林和张检验统计量的循环化版本。考虑两种具体循环化方式：一种通过取相应扫描检验统计量的平均值，另一种则采用最大值。在一定程度上，此循环化技术有效消除了位置效应，使权重能够聚焦于不同信号频率。针对有限样本性能的模拟研究（虽有限但具说服力）表明，循环化张方法优于循环化安德森-达林，且循环化检验优于其母体方法。同时获得平均型循环化的大样本理论结果：循环化安德森-达林与循环化张的渐近分布均为无限个独立标准正态随机变量平方的加权和。此外，核矩阵与核函数均为循环矩阵/函数，因此可通过快速傅里叶变换实现高效的渐近近似计算。