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 X X with distribution function F X {F}_{X} is a uniformly distributed random variable U = F X ( X ) U={F}_{X}\left(X) . We define the angular probability integral transform (APIT) as θ U = 2 π U = 2 π F X ( 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 2 π \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 2 π \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, X 1 {X}_{1} and X 2 {X}_{2} , and test for the circular uniformity of their sum (difference) modulus 2 π \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.