Pivotal Bootstrap Methods for k-Sample Problems in Directional Statistics and Shape Analysis

Abstract

We propose a novel bootstrap hypothesis testing approach for the problem of testing a null hypothesis of a common mean direction, mean polar axis, or mean shape across several populations of real unit vectors (the directional case) or complex unit vectors (the two-dimensional shape case). Multisample testing problems of this type arise frequently in directional statistics and shape analysis (as in other areas of statistics), but to date there has been relatively little discussion of nonparametric bootstrap approaches to this problem. The bootstrap approach described here is based on a statistic that can be expressed as the smallest eigenvalue of a certain positive definite matrix. We prove that this statistic has a limiting chi-squared distribution under the null hypothesis of equality of means across populations. Although we focus mainly on the version of the statistic in which neither isotropy within populations nor constant dispersion structure across populations is assumed, we explain how to modify the statistic so that either or both of these assumptions can be incorporated. Our numerical results indicate that the bootstrap approach proposed here may be expected to perform well in practice.

KEY WORDS:

• Bootstrap hypothesis testing
• Mean direction
• Mean polar axis
• Mean shape
• Permutation test
• Preshape