Open Access
April 2001 Tail probabilities of the maxima of multilinear forms and their applications
Satoshi Kuriki, Akimichi Takemura
Ann. Statist. 29(2): 328-371 (April 2001). DOI: 10.1214/aos/1009210545


Let Z be a k­way array consisting of independent standard normal variables. For column vectors h1, …, hk, define a multilinear form of degree k by $(h_1 \otimes \cdots \otimes h_k)' \vec(Z)$. We derive formulas for upper tail probabilities of the maximum of a multilinear form with respect to the hi’s under the condition that the hi’s are unit vectors, and of its standardized statistic obtained by dividing by the norm of Z. We also give formulas for the maximum of a symmetric multilinear form $(h_1 \otimes \cdots \otimes h_k)' \sym(Z)$, where sym (Z) denotes the symmetrization of Z with respect to indices. These classes of statistics are used for testing hypotheses in the analysis of variance of multiway layout data and for testing multivariate normality. In order to derive the tail probabilities we employ a geometric approach developed by Hotelling, Weyl and Sun. Upper and lower bounds for the tail probabilities are given by reexamining Sun's results. Some numerical examples are given to illustrate the practical usefulness of the obtained formulas, including the upper and lower bounds.


Download Citation

Satoshi Kuriki. Akimichi Takemura. "Tail probabilities of the maxima of multilinear forms and their applications." Ann. Statist. 29 (2) 328 - 371, April 2001.


Published: April 2001
First available in Project Euclid: 24 December 2001

zbMATH: 1103.62351
MathSciNet: MR1863962
Digital Object Identifier: 10.1214/aos/1009210545

Primary: 62H10 , 62H15
Secondary: 53C65

Keywords: empirical orthogonal functions , Gaussian field , Karhunen-Loève expansion , Largest eigenvalue , Multiple comparisons , multivariate normality , multiway layout , PARAFAC , Projection pursuit , tube formula , Wishart distribution

Rights: Copyright © 2001 Institute of Mathematical Statistics

Vol.29 • No. 2 • April 2001
Back to Top