The Volume-of-Tubes Formula: Computational Methods and Statistical Applications

TL;DR

This paper discusses the development of computational methods and software for applying the volume-of-tube formula across various statistical inference problems, enhancing its practical utility.

Contribution

It introduces the libtube software library for numerical evaluation of geometric constants in the volume-of-tube formula, facilitating its application in diverse statistical contexts.

Findings

libtube enables efficient computation of geometric constants

Applications include confidence bands, functional data analysis, and spatial scan analysis

Illustrative examples demonstrate practical use cases

Abstract

The volume-of-tube formula was first introduced by Hotelling (1939), to solve significance of terms in nonlinear regression models. Since this pioneering paper, there has been significant work on extending the tube formula to more general settings, including multidimensional problems, and many new applications in statistical inference, including confidence bands in regression and smoothing models; applications to functional data analysis; testing in mixture models; and spatial scan analysis. Implementation of the tube formula requires numerical evaluation of certain problem-specific geometric constants that appear in Hotelling's formula and its extensions. The purpose of this note is to describe a software library, libtube, that performs the calculations. A variety of illustrative examples are given. Source code for the libtube library andexamples can be downloaded from http://www.herine.net/stat/libtube/.