A useful decomposition of the resultant length for samples from von Mises-Fisher distributions

Abstract

Let x₁, x₂, … be a sequence of independent random vectorsfrom the p-variate von Mises-Fisher distribution with zero meandirection and concentration parameter K, and let R_mresultant length of x₁, …, x_m. In the decomposition , the two components are approximately independently distributed as x²-variates with respectivedegrees of freedom (p−1)(n−t−1) and (p−l)t, provided that k is of moderate size and that t is moderately small relative to n. Hence, is approximately distributed as an F-variate with degrees of freedom (p−l)t and (p−1)(n−t−1). Simulation studies for the cases p=2 and p=3, with t=l, suggest that this last approximation may be reasonable for k as low as 2.5, and n=5,6,… . Some recommendations for larger values of t are made, and an application to the theory of outliers for samples from von Mises-Fisher distributions is mentioned.

Keywords:

• Von Mises-Fisher distribution
• directional distributions
• resultant length