Test of bivariate independence based on angular probability integral transform with emphasis on circular-circular and circular-linear data

The probability integral transform of a continuous random variable XX with distribution function FX{F}_{X} is a uniformly distributed random variable U=FX(X)U={F}_{X}\left(X). We define the angular probability integral transform (APIT) as θU=2πU=2πFX(X){\theta }_{U}=2\pi U=2\pi {F}_{X}\left(X), which corresponds to a uniformly distributed angle on the unit circle. For circular (angular) random variables, the sum modulus 2π\pi of absolutely continuous independent circular uniform random variables is a circular uniform random variable, that is, the circular uniform distribution is closed under summation modulus 2π\pi , and it is a stable continuous distribution on the unit circle. If we consider the sum (difference) of the APITs of two random variables, X1{X}_{1} and X2{X}_{2}, and test for the circular uniformity of their sum (difference) modulus 2π\pi , this is equivalent to 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 by generating samples from NNTS alternative distributions that could be at a closer proximity with respect to the circular uniform null distribution.