The theory of concentrated Langevin distributions

AbstractThe density of the Langevin (or Fisher-Von Mises) distribution is proportional to exp κμ′x, where x and the modal vector μ are unit vectors in Rq. κ (≥0) is called the concentration parameter. The distribution of statistics for testing hypotheses about the modal vectors of m distributions simplify greatly as the concentration parameters tend to infinity. The non-null distributions are obtained for statistics appropriate when κ1,…,κm are known but tend to infinity, and are unknown but equal to κ which tends to infinity. The three null hypotheses are H01:μ = μ0(m=1), H02:μ1 = … =μm, H03:μi ϵ V, i=1,…,m In each case a sequence of alternatives is taken tending to the null hypothesis