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

TL;DR

This paper introduces NNTS-based circular uniformity tests that leverage the closure property of NNTS distributions under summation, providing flexible tools to detect deviations from uniformity on the circle.

Contribution

It develops new NNTS-based tests for circular uniformity using maximum likelihood and likelihood ratio methods, addressing nonregularity issues with simulation-based critical values.

Findings

NNTS distributions are closed under summation.

Proposed tests effectively detect deviations from uniformity.

Critical values are obtained via simulation for various sample sizes.

Abstract

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.