The Annals of Statistics

Weights of $overline{\chi}{}\sp 2$ distribution for smooth or piecewise smooth cone alternatives

Satoshi Kuriki and Akimichi Takemura

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We study the problem of testing a simple null hypothesis about the multivariate normal mean vector against smooth or piecewise smooth cone alternatives. We show that the mixture weights of the $\bar{\chi}^2$ distribution of the likelihood ratio test can be characterized as mixed volumes of the cone and its dual. The weights can be calculated by integration involving the second fundamental form on the boundary of the cone. We illustrate our technique by examples involving a spherical cone and a piecewise smooth cone.

Article information

Ann. Statist., Volume 25, Number 6 (1997), 2368-2387.

First available in Project Euclid: 30 August 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H10: Distribution of statistics 62H15: Hypothesis testing
Secondary: 52A39: Mixed volumes and related topics

Multivariate one-sided alternative one-sided simultaneous confidence region mixed volume second fundamental form volume element internal angle external angle Gauss-Bonnet theorem Shapiro's conjecture


Takemura, Akimichi; Kuriki, Satoshi. Weights of $overline{\chi}{}\sp 2$ distribution for smooth or piecewise smooth cone alternatives. Ann. Statist. 25 (1997), no. 6, 2368--2387. doi:10.1214/aos/1030741077.

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