独立圆形随机变量之和以及基于非负三角和分布的最大似然圆形均匀性检验
Sums of Independent Circular Random Variables and Maximum Likelihood Circular Uniformity Tests Based on Nonnegative Trigonometric Sums Distributions
AI总结：
本文证明 NNTS 圆形分布族在求和下封闭，并针对均匀分布位于参数空间边界导致的非正则性，提出两种基于最大似然的圆形均匀性检验，通过模拟与回归插值确定临界值并验证其有效性。
摘要：
单位圆上的圆形均匀分布在求和运算下是封闭的，也就是说，独立的圆形均匀分布随机变量之和同样服从圆形均匀分布。本研究表明，一类基于非负三角和（NNTS）的圆形分布在求和运算下也是封闭的。鉴于 NNTS 圆形分布在建模多峰性和偏度方面的灵活性，它们非常适合作为替代模型，用于检验圆形均匀性，以检测偏离圆形均匀性原假设的各种不同情形。圆形均匀分布是 NNTS 分布族的一员，但在 NNTS 参数空间中，它对应于参数空间边界上的一个点，这意味着在使用最大似然方法估计参数时，正则性条件无法得到满足。通过考虑标准化最大似然估计量和广义似然比，本文提出了两种 NNTS 圆形均匀性检验。鉴于非正则性条件，所提出的 NNTS 圆形均匀性检验的临界值通过模拟获得，并通过拟合回归模型对任意样本量进行插值。最后，通过生成接近圆形均匀性原假设的 NNTS 模型，评估了所提出的 NNTS 圆形均匀性检验的有效性。
英文摘要：
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.
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