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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.

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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[1] Anderson, T. W. (1984). An Introduction to Multivariate Statistical Analysis, 2nd ed. Wiley, New York. · Zbl 0651.62041
[2] Araujo, A. and Giné, E. (1980). The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, New York. · Zbl 0457.60001
[3] Baringhaus, L. and Henze, N. (1991). Limit distributions for measures of multivariate skewness and kurtosis based on projections. J. Multivariate Anal. 38 51-69. · Zbl 0728.62023
[4] Boik, R. J. (1990). A likelihood ratio test for three-mode singular values: upper percentiles and an application to three-way ANOVA. Comput. Statist. Data Anal. 10 1-9. · Zbl 0825.62379
[5] Boik, R. J. and Marasinghe, M. G. (1989). Analysis of nonadditive multiway classifications. J. Amer. Statist. Assoc. 84 1059-1064. · Zbl 0721.62057
[6] Davis, A. W. (1972). On the ratios of the individual latent roots to the trace of a Wishart matrix. J. Multivariate Anal. 2 440-443. · Zbl 0252.62029
[7] Friedman, J. H. (1987). Exploratory projection pursuit. J. Amer. Statist. Assoc. 82 249-266. JSTOR: · Zbl 0664.62060
[8] Gray, A. (1990). Tubes. Addison-Wesley, Redwood City, CA. · Zbl 0692.53001
[9] Hanumara, R. C. and Thompson, Jr., W. A. (1968). Percentage points of the extreme roots of a Wishart matrix. Biometrika 55 505-512. JSTOR: · Zbl 0177.46705
[10] Hotelling, H. (1939). Tubes and spheres in n-spaces, and a class of statistical problems. Amer. J. Math. 61 440-460. JSTOR: · Zbl 0020.38302
[11] Huber, P. J. (1985). Projection pursuit. Ann. Statist. 13 435-475. · Zbl 0595.62059
[12] Johansen, S. and Johnstone, I. (1990). Hotelling’s theorem on the volume of tubes: some illustrations in simultaneous inference and data analysis. Ann. Statist. 18 652-684. · Zbl 0723.62018
[13] Johnson, D. E. and Graybill, F. A. (1972). An analysis of a two-way model with interaction and no replication. J. Amer. Statist. Assoc. 67 862-868. JSTOR: · Zbl 0254.62042
[14] Kawasaki, H. and Miyakawa, M. (1996). A test of three-factor interaction in a three-way layout without replication. J. Japanese Society for Quality Control 26 97-108 (in Japanese).
[15] Knowles, M. and Siegmund, D. (1989). On Hotelling’s approach to testing for a nonlinear parameter in regression. Internat. Statist. Rev. 57 205-220. · Zbl 0707.62125
[16] Kuriki, S. and Takemura, A. (2000). Shrinkage estimation towards a closed convex set with a smooth boundary. J. Multivariate Anal. 75 79-111. · Zbl 0983.62033
[17] Leurgans, S. and Ross, R. T. (1992). Multilinear models: applications in spectroscopy (with discussion). Statist. Sci. 7 289-319. · Zbl 0955.62592
[18] Machado, S. G. (1983). Two statistics for testing for multivariate normality. Biometrika 70 713-718. JSTOR: · Zbl 0543.62010
[19] Malkovich, J. F. and Afifi, A. A. (1973). On tests for multivariate normality. J. Amer. Statist. Assoc. 68 176-179.
[20] Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York. · Zbl 0556.62028
[21] Pillai, K. C. S. (1976). Distributions of the characteristic roots in multivariate analysis I. Null distribution. Canad. J. Statist. Sec. A and B 4 157-184. JSTOR: · Zbl 0378.62055
[22] Schuurmann, F. J., Krishnaiah, P. R. and Chattopadhyay, A. K. (1973). On the distribution of the ratios of the extreme roots to the trace of the Wishart matrix. J. Multivariate Anal. 3 445-453. · Zbl 0286.62031
[23] Sun, J. (1991). Significance levels in exploratory projection pursuit. Biometrika 78 759-769. JSTOR: · Zbl 0753.62067
[24] Sun, J. (1993). Tail probabilities of the maxima of Gaussian random fields. Ann. Probab. 21 34-71. · Zbl 0772.60038
[25] Takemura, A. (1993). Maximally orthogonally invariant higher order moments and their application to testing elliptically-contouredness. In Statistical Science and Data Analysis (K. Matushita, M. L. Puri and T. Hayakawa, eds.) 225-235. VSP, Utrecht. · Zbl 0897.62054
[26] Takemura, A. and Kuriki, S. (1997). Weights of \? 2 distribution for smooth or piecewise smooth cone alternatives. Ann. Statist. 25 2368-2387. · Zbl 0897.62055
[27] Weyl, H. (1939). On the volume of tubes. Amer. J. Math. 61 461-472. JSTOR: · Zbl 0021.35503
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