## The Annals of Statistics

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

#### Abstract

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

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

Dates
First available in Project Euclid: 30 August 2002

https://projecteuclid.org/euclid.aos/1030741077

Digital Object Identifier
doi:10.1214/aos/1030741077

Mathematical Reviews number (MathSciNet)
MR1604465

Zentralblatt MATH identifier
0897.62055

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

#### Citation

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. https://projecteuclid.org/euclid.aos/1030741077

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