Large sample theory of the Langevin distribution

The Langevin (or von Mises-Fisher) distribution of random vector x on the unit sphere ω_q in $\text{R}$^q has a density proportional to exp κμ'x where μ'x is the scalar product of x with the unit modal vector μ and κ?0 is a concentration parameter. This paper studies estimation and tests for a wide variety of situations when the sample sizes are large. Geometrically simple test statistics are given for many sample problems even when the populations may have unequal concentration parameters.