No. 1043

Title:

(originally in Japanese)
The volume of tubes and the distribution of the maximum of a Gaussian random field

Author(s):

Kuriki, Satoshi (Institute of Statistical Mathematics)
Takemura, Akimichi (University of Tokyo)

Key words:

Euler characteristic heuristics; Morse's theorem

Abstract:

Contents
1. Introduction

1.1 Gaussian random fields on a manifold
1.2 The tube method and the Euler characteristic heuristics

2. The tube method

2.1 The volume of tubes and the distribution of the maximum
2.2 Evaluation of the volume of tubes
2.3 Evaluation of the critical radii
2.4 Evaluation of the precision of the tube method
2.5 The case of closed manifolds

3. The Euler characteristic heuristics

3.1 The excursion set
3.2 Morse's theorem and the approximation formula
3.3 The case of Gaussian random fields
3.4 Evaluation of the precision of the Euler characteristic heuristics

4. Applications to statistics

4.1 The maximum of the multilinear forms and the multi-way ANOVA
4.2 Significance tests in projection pursuit