Hotelling's Theorem on the Volume of Tubes: Some Illustrations in Simultaneous Inference and Data Analysis

Abstract

We illustrate with contemporary examples Hotelling's geometric approach to simultaneous probability calculations. Hotelling reduces the evaluation of certain normal theory significance probabilities to finding the volume of a tube about a curve in a hypersphere, and shows that this volume is often exactly given by length times cross-sectional area. We review Hotelling's result together with some recent complements, and then use the approach to set simultaneous prediction regions for some data from gait analysis, to study Andrews' plots in multivariate data analysis, and to construct significance tests for projection pursuit regression. A by-product is a numerical criterion for tube self-overlap, relevant, for example, to uniqueness of certain nonlinear least squares estimates.