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The circular uniform distribution on the unit circle is closed undersummation, that is, the sum of independent circular uniformly distributedrandom variables is also circular uniformly distributed. In this study, it isshown that a family of circular distributions based on nonnegativetrigonometric sums (NNTS) is also closed under summation. Given the flexibilityof NNTS circular distributions to model multimodality and skewness, these aregood candidates for use as alternative models to test for circular uniformityto detect different deviations from the null hypothesis of circular uniformity.The circular uniform distribution is a member of the NNTS family, but in theNNTS parameter space, it corresponds to a point on the boundary of theparameter space, implying that the regularity conditions are not satisfied whenthe parameters are estimated by using the maximum likelihood method. Two NNTStests for circular uniformity were developed by considering the standardisedmaximum likelihood estimator and the generalised likelihood ratio. Given thenonregularity condition, the critical values of the proposed NNTS circularuniformity tests were obtained via simulation and interpolated for any samplesize by the fitting of regression models. The validity of the proposed NNTScircular uniformity tests was evaluated by generating NNTS models close to thecircular uniformity null hypothesis.

Sums of Independent Circular Random Variables and Maximum Likelihood Circular Uniformity Tests Based on Nonnegative Trigonometric Sums Distributions