The probability integral transform (PIT) of a continuous random variable $X$ with distribution function $F_X$ is a uniformly distributed random variable $U=F_X(X)$. We define the angular probability integral transform (APIT) as $\theta_U = 2 \pi U = 2 \pi F_{X}(X)$, which corresponds to a uniformly distributed angle on the unit circle. For circular (angular) random variables, the sum of absolutely continuous independent circular uniform random variables is a circular uniform random variable, that is, the circular uniform distribution is closed under summation, and it is a stable continuous distribution on the unit circle. If we consider the sum (difference) of the angular probability integral transforms of two random variables, $X_1$ and $X_2$, and test for the circular uniformity of their sum (difference), this is equivalent to the test of independence of the original variables. In this study, we used a flexible family of nonnegative trigonometric sums (NNTS) circular distributions, which include the uniform circular distribution as a member of the family, to evaluate the power of the proposed independence test; we complete this evaluation by generating samples from NNTS alternative distributions that may be at a closer proximity with respect to the circular uniform null distribution.

翻译：概率积分变换（PIT）将具有分布函数 $F_X$ 的连续随机变量 $X$ 转换为均匀分布随机变量 $U=F_X(X)$。我们定义角概率积分变换（APIT）为 $\theta_U = 2 \pi U = 2 \pi F_{X}(X)$，其对应单位圆上的均匀分布角度。对于圆形（角度）随机变量，绝对连续独立圆形均匀随机变量之和仍为圆形均匀随机变量，即圆形均匀分布在加法下封闭，且为单位圆上的稳定连续分布。若考虑两个随机变量 $X_1$ 与 $X_2$ 的角概率积分变换之和（差），并通过检验其和（差）的圆形均匀性来等价于原始变量的独立性检验。本研究采用包含均匀圆形分布作为成员的灵活非负三角和（NNTS）圆形分布族，评估所提独立性检验的统计功效；通过从与圆形均匀零假设分布距离较近的NNTS备择分布中生成样本，完成此项评估。