Multimodal Symmetric Circular Distributions Based on Nonnegative Trigonometric Sums and a Likelihood Ratio Test for Reflective Symmetry

Abstract: Fern\'andez-Dur\'an (2004) developed a family of circular distributions based on nonnegative trigonometric sums (NNTS) which is flexible for modeling datasets exhibiting multimodality and asymmetry. Many datasets involving angles in the natural sciences, such as animal movement in biology, are expected to exhibit reflective symmetry with respect to a central angle (axis) of symmetry. Testing for symmetry in the underlying circular density from which these angles are generated is crucial. Additionally, such densities often display multimodality. This paper identifies the conditions under which NNTS distributions are reflective symmetric and develops a likelihood ratio test for reflective symmetry. The proposed methodology is demonstrated through applications to simulated and real datasets.