Nonparametric smoothing of directional and axial data

We discuss generalized linear models for directional data where the conditional distribution of the response is a von Mises–Fisher distribution in arbitrary dimension or a Bingham distribution on the unit circle. To do this properly, we parametrize von Mises–Fisher distributions by Euclidean parameters and investigate computational aspects of this parametrization. Then we modify this approach for local polynomial regression as a means of nonparametric smoothing of distributional data. The methods are illustrated with simulated data and a dataset from planetary sciences involving covariate vectors on a sphere with axial response.