Annals of Statistics

Random fields of multivariate test statistics, with applications to shape analysis

J. E. Taylor and K. J. Worsley

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Our data are random fields of multivariate Gaussian observations, and we fit a multivariate linear model with common design matrix at each point. We are interested in detecting those points where some of the coefficients are nonzero using classical multivariate statistics evaluated at each point. The problem is to find the P-value of the maximum of such a random field of test statistics. We approximate this by the expected Euler characteristic of the excursion set. Our main result is a very simple method for calculating this, which not only gives us the previous result of Cao and Worsley [Ann. Statist. 27 (1999) 925–942] for Hotelling’s T2, but also random fields of Roy’s maximum root, maximum canonical correlations [Ann. Appl. Probab. 9 (1999) 1021–1057], multilinear forms [Ann. Statist. 29 (2001) 328–371], χ̄2 [Statist. Probab. Lett 32 (1997) 367–376, Ann. Statist. 25 (1997) 2368–2387] and χ2 scale space [Adv. in Appl. Probab. 33 (2001) 773–793]. The trick involves approaching the problem from the point of view of Roy’s union-intersection principle. The results are applied to a problem in shape analysis where we look for brain damage due to nonmissile trauma.

Article information

Ann. Statist., Volume 36, Number 1 (2008), 1-27.

First available in Project Euclid: 1 February 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 62H11: Directional data; spatial statistics

Integral geometry differential topology Euler characteristic Morse theory Roy’s maximum root Hotelling’s T^2 canonical correlation scale space


Taylor, J. E.; Worsley, K. J. Random fields of multivariate test statistics, with applications to shape analysis. Ann. Statist. 36 (2008), no. 1, 1--27. doi:10.1214/009053607000000406.

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