CIRJE-F-4. Takemura, Akimichi, "Tail Probabilities of the Maxima of Multilinear Forms and Their Applications", June 1998.

Let Z be a k-way array whose q1 x...x qk elements are independent standard normal variables. For qi-dimensional vector hi, i=1, ...., k, define a multilinear form of degree k by (h1 x hk)'vec(Z). We derive formulas for upper tail probabilities of the maximum of multilinear form with respect to hi's under the condition ||hi||=1 for any i, and of its standardized statistic obtained by dividing by ||vec(Z)||. We also give formulas for the maximum of symmetric multilinear form (h1 x...x hk)'vec(sym(Z)),�@where sym(Z) denotes the symmetrization of Z with respect to indices. These classes of statistics have important applications in testing hypotheses of multivariate analysis such as the analysis of variance of multiway layout data or testing multivariate normality. In order to derive the tail probabilities we employ a geometric approach developed by H. Weyl and J. Sun. Upper and lower bounds for the tail probabilities are given by reexamining the Sun's results. Some numerical examples are given to illustrate the practical usefulness of the obtained formulas including the upper and lower bounds.