Application of tube formula to distributional problems in multiway layouts

Recently, an integral geometric method called the tube method has been actively developed. The tube method gives us a powerful tool to tackle problems where conventional matrix theory such as the singular value decomposition cannot be applied. The aim of this paper is to survey several recent applications of the tube method to distributional problems in multiway layouts mainly by the authors. Null distributions of test statistics of the following four testing problems are discussed: (i) A test for interaction in three-way layout based on the three-way analogue of the largest singular value. (ii) A test for multivariate normality by searching a nonnormal direction proposed by Malkovich and Afifi. (iii) Testing independence in ordered categorical data by maximizing row and column scores under order restriction. (iv) Detecting a change point in two-way layout with ordinal factors.