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

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 Rmresultant length of x1, …, xm. 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.