## Annals of Statistics

### Tail probabilities of the maxima of multilinear forms and their applications

#### Abstract

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

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

Dates
First available in Project Euclid: 24 December 2001

https://projecteuclid.org/euclid.aos/1009210545

Digital Object Identifier
doi:10.1214/aos/1009210545

Mathematical Reviews number (MathSciNet)
MR1863962

Zentralblatt MATH identifier
1103.62351

#### Citation

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. https://projecteuclid.org/euclid.aos/1009210545

#### References

• Anderson, T. W. (1984). An Introduction to Multivariate Statistical Analysis, 2nd ed. Wiley, New York.
• Araujo, A. and Gin´e, E. (1980). The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, New York.
• Baringhaus, L. and Henze, N. (1991). Limit distributions for measures of multivariate skewness and kurtosis based on projections. J. Multivariate Anal. 38 51-69.
• 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.
• Boik, R. J. and Marasinghe, M. G. (1989). Analysis of nonadditive multiway classifications. J. Amer. Statist. Assoc. 84 1059-1064.
• 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.
• Friedman, J. H. (1987). Exploratory projection pursuit. J. Amer. Statist. Assoc. 82 249-266.
• Gray, A. (1990). Tubes. Addison-Wesley, Redwood City, CA.
• Hanumara, R. C. and Thompson, Jr., W. A. (1968). Percentage points of the extreme roots of a Wishart matrix. Biometrika 55 505-512.
• Hotelling, H. (1939). Tubes and spheres in n-spaces, and a class of statistical problems. Amer. J. Math. 61 440-460.
• Huber, P. J. (1985). Projection pursuit. Ann. Statist. 13 435-475.
• 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.
• 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.
• 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).
• Knowles, M. and Siegmund, D. (1989). On Hotelling's approach to testing for a nonlinear parameter in regression. Internat. Statist. Rev. 57 205-220.
• Kuriki, S. and Takemura, A. (2000). Shrinkage estimation towards a closed convex set with a smooth boundary. J. Multivariate Anal. 75 79-111.
• Leurgans, S. and Ross, R. T. (1992). Multilinear models: applications in spectroscopy (with discussion). Statist. Sci. 7 289-319.
• Machado, S. G. (1983). Two statistics for testing for multivariate normality. Biometrika 70 713-718.
• Malkovich, J. F. and Afifi, A. A. (1973). On tests for multivariate normality. J. Amer. Statist. Assoc. 68 176-179.
• Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York.
• 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.
• 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.
• Sun, J. (1991). Significance levels in exploratory projection pursuit. Biometrika 78 759-769.
• Sun, J. (1993). Tail probabilities of the maxima of Gaussian random fields. Ann. Probab. 21 34-71.
• 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.
• Takemura, A. and Kuriki, S. (1997). Weights of ¯ 2 distribution for smooth or piecewise smooth cone alternatives. Ann. Statist. 25 2368-2387.
• Weyl, H. (1939). On the volume of tubes. Amer. J. Math. 61 461-472.