Maximum Likelihood Characterization of the Von Mises Distribution

In 1918 von Mises discovered the circular probability distribution which now bears his name by studying the analogue on the circle of the Gaussian maximum likelihood characterization of the normal distribution on the line. Teicher [6] gave a specific condition on the density function for the Gaussian characterization to hold (see also Kagan, et al. [2], p, 411). It is shown in this paper that the same condition is sufficient for the von Mises characterization to be valid. The proof is based on Teicher’s approach, but the periodic nature of circular densities makes substantial modifications necessary. An extension to the hyperspherical case is also given.