Test of Bivariate Independence Based on Angular Probability Integral Transform with Emphasis on Circular-Circular and Circular-Linear Data

The probability integral transform (PIT) of a 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 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.