Large sample theory of the Langevin distribution (Q794082)

A random vector x on a unit sphere \(S_ q\) is said to have the Langevin (or von Mises-Fisher) distribution, if the density function with respect to Lebesgue measure on \(S_ q\) is given by \(a_ q(\kappa)^{- 1}\exp(\kappa \mu^ tx)\) where \(\mu^ t\mu =\| \mu \|^ 2=1\) and \(\kappa\geq 0\). \(\mu\) is the modal vector and \(\kappa\) is a concentration parameter. In this paper estimation and testing problems for \(\mu\) and \(\kappa\) are discussed. Asymptotic non-null distributions of test statistics are given and methods of checking the goodness of fit of the Langevin distribution for large samples are demonstrated.