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.
Publié le : 1997-12-14
Classification:
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,
62H10,
62H15,
52A39
@article{1030741077,
author = {Takemura, Akimichi and Kuriki, Satoshi},
title = {Weights of $overline{\chi}{}\sp 2$ distribution for smooth or piecewise smooth cone alternatives},
journal = {Ann. Statist.},
volume = {25},
number = {6},
year = {1997},
pages = { 2368-2387},
language = {en},
url = {http://dml.mathdoc.fr/item/1030741077}
}
Takemura, Akimichi; Kuriki, Satoshi. Weights of $overline{\chi}{}\sp 2$ distribution for smooth or piecewise smooth cone alternatives. Ann. Statist., Tome 25 (1997) no. 6, pp. 2368-2387. http://gdmltest.u-ga.fr/item/1030741077/