Efficient hypothesis testing for principal orientation in circular von Mises-Based rotational distributions

In this article, we develop a one-sample testing procedure for the principal orientation matrix S, a 3×3 orthogonal matrix that describes the dominant directional tendency of objects in R3 and plays a central role in directional statistics. We propose the Integrated Likelihood Ratio Test (ILRT) as a robust alternative to the classical Likelihood Ratio Test (LRT) for inference on S. To address challenges arising from small sample sizes and low concentration levels, we further refine the procedure using the Computational Approach Test (CAT). The performance of the proposed methods is evaluated through extensive simulation studies, demonstrating that CAT substantially improves accuracy in terms of both size and power, particularly in scenarios with small samples and low concentrations.