  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 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 the 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;
we complete this evaluation by generating samples from NNTS alternative
distributions that may be at a closer proximity with respect to the circular
uniform null distribution.
