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The problem of classification of an observation into von Mises-Fisher populations is considered when the concentration parameters and the mean directions are unknown. For two von Mises-Fisher distributions, we propose the restricted maximum likelihood estimators (MLEs), and Bayes estimators of the concentration parameters. The MLEs and restricted MLEs of the concentration parameters are compared in terms of risks. When the concentration parameters are ordered, we propose classification rules using the restricted MLEs and Bayes estimators of the parameters. For two populations, we also derive predictive Bayes classification rules using informative priors for the concentration parameters. We derive the likelihood ratio-based classification rule. Nonparametric rules such as  k -NN rule, support vector machine classifier, and kernel density-based rule are also studied. For more than two populations, we suggest classification rules. Comparisons among the proposed rules have been carried out with respect to the expected probability of misclassification. Applications of the rules are described using directional data sets.

Classification of observations into von Mises-Fisher populations with unknown parameters