Bootstrap Confidence Regions for Directional Data

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Abstract
Methods are proposed for constructing bootstrap confidence regions for the mean direction of a random p-dimensional unit vector X with an arbitrary unimodal distribution on the p sphere. The approach of this article differs from that of other authors in that it is based on pivotal statistics. A general pivotal method is introduced that produces a wide variety of confidence regions on general p-dimensional spheres; included are confidence cones and likelihood-based regions. It can readily be modified to incorporate extra assumptions about the underlying distribution, such as rotational symmetry. The general method leads to confidence pictures, which present information about the estimated posterior likelihood of mean orientation by shading spherical surfaces. An application is given to a sample of spherical cross-bed measurements. The methods extend to the case where X has random length, and to calculation of confidence regions for reference directions of axial bipolar or girdle distributions.

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