Spherical Regression for Concentrated Fisher-Von Mises Distributions

Abstract

Spherical regression studies models which postulate that the unit vector $v$ is equal to an unknown rotation $P$ of the unit vector $u$ "plus" an experimental error. The case where the experimental errors follow a Fisher-von Mises distribution with a large concentration parameter $\kappa$ is considered in this work. Asymptotic $(\kappa \rightarrow \infty)$ inferential procedures for $P$ are proposed when $n$, the sample size, is fixed. Diagnostic methods for spherical regression are suggested. The key for their derivation is the fact that spherical regression is "locally" identical to ordinary least square regression. The results are presented in an arbitrary dimension. For the three-dimensional case, asymptotic tests and confidence regions for the axis and the angle of $P$ are obtained. The data from a plate tectonic analysis of the Gulf of Aden, presented by Cochran, illustrate the proposed methodology.