The Annals of Statistics

Tail probabilities of the maxima of multilinear forms and their applications

Satoshi Kuriki and Akimichi Takemura

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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.

Article information

Ann. Statist., Volume 29, Number 2 (2001), 328-371.

First available in Project Euclid: 24 December 2001

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Zentralblatt MATH identifier

Primary: 62H10: Distribution of statistics 62H15: Hypothesis testing
Secondary: 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]

Gaussian field Karhunen-Loève expansion empirical orthogonal functions largest eigenvalue multiple comparisons multivariate normality multiway layout PARAFAC projection pursuit tube formula Wishart distribution


Kuriki, Satoshi; Takemura, Akimichi. Tail probabilities of the maxima of multilinear forms and their applications. Ann. Statist. 29 (2001), no. 2, 328--371. doi:10.1214/aos/1009210545.

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