Wrapped Cauchy Robust Approach to the Circular-Circular Regression Model

Circular–circular regression models are widely used to investigate relationships between angular variables in various applied fields, including biostatistics. The classical von Mises (vM) circular–circular regression model, however, is known to be sensitive to outliers due to its light-tailed error structure. In this study, we investigate the wrapped Cauchy (WC) circular–circular regression model as a robust alternative to the vM-based approach for analyzing circular data contaminated by outliers. Parameter estimation is performed via maximum likelihood (ML) using a modern constrained gradient-based optimization algorithm, namely the limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm with box constraints (L-BFGS-B), allowing for stable estimation under natural parameter bounds. Extensive simulation studies demonstrate that, under contaminated settings, the WC model provides substantially more stable parameter estimates than the vM model, yielding markedly lower mean squared error and variability, particularly for high concentration regimes and directional outliers. The robustness advantage of the WC model is further illustrated through a real biostatistical application involving the circular relationship between the months of diagnosis and surgical intervention in gastric cancer patients. Overall, the results highlight the practical benefits of WC-based circular–circular regression for robust inference in the presence of outliers.