Scaled von Mises-Fisher distributions

The best-known distribution on the sphere for statistical purposes is the Fisher, or  von Mises-Fisher, distribution.  However, this distribution possesses rotational symmetry which is too strong an assumption for many applications.  Here we present a new distribution, called the scaled von Mises-Fisher (SvMF) distribution, which has ellipse-like contours yet has a simple normalising constant.   The SvMF distribution will be compared and contrasted with two other distributions on the sphere with ellipse-like symmetry, the Kent and the elliptically symmetric angular Gaussian distributions.  A main goal for developing this distribution is to use it for regression modelling of paleomagnetic data with a directional response.  Applications to such data will be described.  Different link functions will be discussed briefly.