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The probability integral transform of a continuous random variableXXwith distribution functionFX{F}_{X}is a uniformly distributed random variableU=FX(X)U={F}_{X}\left(X). We define the angular probability integral transform (APIT) asθU=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\piof 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}andX2{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.

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