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 which are of practical importance but hardly handled by conventional methods. Null distributions of test statistics of the following three 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) Testing independence in ordered categorical data by maximizing row and column scores under order restriction. (iii) Detecting a change point in two-way layout with ordinal factors. Copyright (C) 2002 John Wiley Sons, Ltd.