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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 uniformlydistributed angle on the unit circle. For circular (angular) random variables,the sum of absolutely continuous independent circular uniform random variablesis a circular uniform random variable, that is, the circular uniformdistribution is closed under summation, and it is a stable continuousdistribution on the unit circle. If we consider the sum (difference) of theangular probability integral transforms of two random variables, $X_1$ and$X_2$, and test for the circular uniformity of their sum (difference), this isequivalent to the test of independence of the original variables. In thisstudy, we used a flexible family of nonnegative trigonometric sums (NNTS)circular distributions, which include the uniform circular distribution as amember of the family, to evaluate the power of the proposed independence test;we complete this evaluation by generating samples from NNTS alternativedistributions that may be at a closer proximity with respect to the circularuniform null distribution.

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