Efficient sampling from the PKBD distribution

In this paper we present and analyze random number generators for the Poisson Kernel-Based Distribution (PKBD) on the sphere. We show that the only currently available sampling scheme presented in Golzy and Markatou (2020) can be improved by a better selection of hyperparameters but still yields an unbounded rejection constant as the concentration parameter approaches 1. Furthermore, we introduce two additional and superior sampling methods for which boundedness in the above mentioned case can be obtained. The first method proposes initial draws from angular central Gaussian distribution and offers uniformly bounded rejection constants for a significant part of the PKBD parameter space. The second method uses adaptive rejection sampling and the results of Ulrich (1984) to sample from the projected Saw distribution (Saw, 1978). Finally, both new methods are compared in a simulation study.