  The circular uniform distribution on the unit circle is closed under
summation, that is, the sum of independent circular uniformly distributed
random variables is also circular uniformly distributed. In this study, it is
shown that a family of circular distributions based on nonnegative
trigonometric sums (NNTS) is also closed under summation. Given the flexibility
of NNTS circular distributions to model multimodality and skewness, these are
good candidates for use as alternative models to test for circular uniformity
to detect different deviations from the null hypothesis of circular uniformity.
The circular uniform distribution is a member of the NNTS family, but in the
NNTS parameter space, it corresponds to a point on the boundary of the
parameter space, implying that the regularity conditions are not satisfied when
the parameters are estimated by using the maximum likelihood method. Two NNTS
tests for circular uniformity were developed by considering the standardised
maximum likelihood estimator and the generalised likelihood ratio. Given the
nonregularity condition, the critical values of the proposed NNTS circular
uniformity tests were obtained via simulation and interpolated for any sample
size by the fitting of regression models. The validity of the proposed NNTS
circular uniformity tests was evaluated by generating NNTS models close to the
circular uniformity null hypothesis.
