Discriminant analysis with high dimensional von Mises-Fisher distributions

This paper extends previous work in discriminant analysis with vMF distributions (e. g., Morris & Laycock, Biometrika, 1974) to general dimension, allowing computation of misclassification probabilities and ROC curves. The key result is the probability distribution of the cosine transformation of a vMF distribution, that is, the random variable Ua = aTX , where X = (X1, ..., Xp)T is a random direction of Sp with vMF distribution and a = (a1, ..., ap)T is a fixed non - random direction of Sp . This transformation is of general importance in multivariate analysis, in particular it underlies discriminant analysis both in the two - group and in the multiple group problem. It allows also to check the surmise that two - group maximum likelihood discriminant rule is equivalent to Fisher's linear discriminant function.