The Annals of Statistics

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

Soren Johansen and Iain M. Johnstone

Full-text: Open access

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.

Article information

Source
Ann. Statist. Volume 18, Number 2 (1990), 652-684.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aos/1176347620

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aos/1176347620

Mathematical Reviews number (MathSciNet)
MR1056331

Zentralblatt MATH identifier
0723.62018

Subjects
Primary: 62E15: Exact distribution theory
Secondary: 62J15: Paired and multiple comparisons 62F25: Tolerance and confidence regions 62J02: General nonlinear regression

Keywords
Volume of tubes simultaneous inference confidence bands prediction bands gait analysis Andrews' plots projection pursuit regression

Citation

Johansen, Soren; Johnstone, Iain M. Hotelling's Theorem on the Volume of Tubes: Some Illustrations in Simultaneous Inference and Data Analysis. The Annals of Statistics 18 (1990), no. 2, 652--684. doi:10.1214/aos/1176347620. http://projecteuclid.org/euclid.aos/1176347620.


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