The tube method for the moment index in projection pursuit

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 [1987. What is projection pursuit? J. Roy. Statist. Soc. Ser. A 150, 1-36]. The 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-Loève 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.