The tube method for the moment index in projection pursuit

TL;DR

This paper analyzes the distribution of the moment index in projection pursuit, deriving an approximation for its tail probability using the tube method and Gaussian random field theory.

Contribution

It introduces a novel application of the tube method to approximate the tail probability of the maximum of the moment index in multivariate analysis.

Findings

Derived the limiting distribution as a Gaussian random field maximum.

Provided an approximate formula for tail probability (p-value).

Applied the tube method to the moment index in projection pursuit.

Abstract

The projection pursuit index defined by a sum of squares of the third and the fourth sample cumulants is known as the moment index proposed by Jones and Sibson. Limiting distribution of the maximum of the moment index under the null hypothesis that the population is multivariate normal is shown to be the maximum of a Gaussian random field with a finite Karhunen-Loeve expansion. An approximate formula for tail probability of the maximum, which corresponds to the p-value, is given by virtue of the tube method through determining Weyl's invariants of all degrees and the critical radius of the index manifold of the Gaussian random field.