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Weights of \(\overline{\chi}^2\) distribution for smooth or piecewise smooth cone alternatives. (English) Zbl 0897.62055

Summary: 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 \(\overline\chi^2\) 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.

MSC:

62H15 Hypothesis testing in multivariate analysis
62H10 Multivariate distribution of statistics
52A39 Mixed volumes and related topics in convex geometry
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