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Tail probabilities of the maxima of multilinear forms and their applications. (English) Zbl 1103.62351
Summary: Let \(Z\) be a \(k\)-way array consisting of independent standard normal variables. For column vectors \(h_1,\dots,h_k\), define a multilinear form of degree \(k\) by \((h_1\otimes\dots\otimes h_k)'\text{vec}(Z)\). We derive formulas for upper tail probabilities of the maximum of a multilinear form with respect to the \(h_i\)’s under the condition that the \(h_i\)’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 \((h_1\otimes\dots\otimes h_k)'\text{vec}(\text{sym}(Z))\), where \(\text{sym}(Z)\) 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 formulas obtained, including the upper and lower bounds.

MSC:
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62G32 Statistics of extreme values; tail inference
62H10 Multivariate distribution of statistics
62H15 Hypothesis testing in multivariate analysis
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