Estimating the concentration parameter of a von Mises distribution: a systematic simulation benchmark

In directional statistics, the von Mises distribution is a key element in the analysis of circular data. While there is a general agreement regarding the estimation of its location parameter <i>μ</i>, several methods have been proposed to estimate the concentration parameter <i>κ</i>. We here provide a thorough evaluation of the behavior of 12 such estimators for datasets of size <i>N</i> ranging from 2 to 8192 generated with a <i>κ</i> ranging from 0 to 100. We provide detailed results as well as a global analysis of the results, showing that (1) for a given <i>κ</i>, most estimators have behaviors that are very similar for large datasets (N≥16) and more variable for small datasets, and (2) for a given estimator, results are very similar if we consider the mean absolute error for κ≤1 and the mean relative absolute error for κ≥1.