Efficient Sampling From the Watson Distribution in Arbitrary Dimensions

In this article, we present two efficient methods for sampling from the Watson distribution in arbitrary dimensions. The first method adapts the rejection sampling algorithm from <i>Kent, Ganeiber, and Mardia</i>, originally designed for Bingham distributions, using angular central Gaussian envelopes. For the Watson distribution, we derive a closed-form expression for the parameters that maximize sampling efficiency, which is further investigated and bounded by asymptotic results. This approach avoids the curse of dimensionality through a smart matrix inversion, enabling fast runtimes even in high dimensions. The second method, based on <i>Saw</i>, employs adaptive rejection sampling from a projected distribution. This algorithm is also effective in all dimensions and offers rapid sampling capabilities. Finally, our simulation study compares the two main methods, revealing that each excels under different conditions: the first method is more efficient for small samples or large dimensions, while the second performs better with larger samples and more concentrated distributions. Both algorithms are available in the R package watson on CRAN. Supplementary materials for this article are available online.