The circular uniform distribution on the unit circle is closed under summation, that is, the sum of independent circular uniformly distributed random variables is also circular uniformly distributed. In this study, it is shown that a family of circular distributions based on nonnegative trigonometric sums (NNTS) is also closed under summation. Given the flexibility of NNTS circular distributions to model multimodality and skewness, these are good candidates for use as alternative models to test for circular uniformity to detect different deviations from the null hypothesis of circular uniformity. The circular uniform distribution is a member of the NNTS family, but in the NNTS parameter space, it corresponds to a point on the boundary of the parameter space, implying that the regularity conditions are not satisfied when the parameters are estimated by using the maximum likelihood method. Two NNTS tests for circular uniformity were developed by considering the standardised maximum likelihood estimator and the generalised likelihood ratio. Given the nonregularity condition, the critical values of the proposed NNTS circular uniformity tests were obtained via simulation and interpolated for any sample size by the fitting of regression models. The validity of the proposed NNTS circular uniformity tests was evaluated by generating NNTS models close to the circular uniformity null hypothesis.

翻译：单位圆上的圆均匀分布在加法下封闭，即独立圆均匀分布随机变量之和仍为圆均匀分布。本研究表明，基于非负三角和（NNTS）的圆分布族同样在加法下封闭。鉴于NNTS圆分布在建模多峰性和偏态性方面的灵活性，它们可作为检验圆均匀性的替代模型，以检测对圆均匀性原假设的不同偏离。圆均匀分布是NNTS族的一个成员，但在NNTS参数空间中，它对应于参数空间边界上的一个点，这意味着当使用最大似然方法估计参数时，正则性条件不满足。通过考虑标准化最大似然估计量和广义似然比，发展了两种用于圆均匀性的NNTS检验。鉴于非正则性条件，通过模拟获得所提NNTS圆均匀性检验的临界值，并利用回归模型拟合以插值出任意样本量下的临界值。通过生成接近圆均匀性原假设的NNTS模型，评估了所提NNTS圆均匀性检验的有效性。