Discriminant Analysis with High Dimensional von Mises - Fisher Distributions

This paper extends previous work in discriminant analysis with von Mises-Fisher distributions (e. g., Morris and Laycock, Biometrika, 1974) to general dimension, allowing computation of misclassification probabilities. The main result is the probability distribution of the cosine transformation of a von Mises-Fisher distribution, that is, the random variable U_a=a^T X, where X=(X_1,…,X_p )^T, satisfying X^T X=1, is a random direction with von Mises-Fisher distribution and a=(a_1,…,a_p )^T, satisfying a^T a=1, is a fixed non-random direction. This transformation is of general interest in multivariate analysis, in particular it underlies discriminant analysis in both two-group and multiple group problem.