Application of tube formula to distributional problems in multiway layouts

Abstract

SUMMARYRecently, 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.

Volume of tubes and distribution of the maxima of Gaussian random fields

“…The Euler characteristic heuristic is more general than the tube method. However, the tube method has many applications to multivariate analysis ( [29], [30], [24]) and it presents more concrete geometric pictures in terms of the tubes. Furthermore, the evaluation of the error term by the tube method can be extended to the Euler characteristic heuristic.…”

“…In Section 4 we present two applications of the tube method to statistics. For other applications, see [30], [11], [25]. Other topics closely related to the tube method and the Euler characteristic heuristic include improvement of joint confidence sets ( [34], [18], [40]) and the discrete tube method ( [35], [36], [37]).…”