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April 2009 Likelihood ratio tests and singularities
Mathias Drton
Ann. Statist. 37(2): 979-1012 (April 2009). DOI: 10.1214/07-AOS571


Many statistical hypotheses can be formulated in terms of polynomial equalities and inequalities in the unknown parameters and thus correspond to semi-algebraic subsets of the parameter space. We consider large sample asymptotics for the likelihood ratio test of such hypotheses in models that satisfy standard probabilistic regularity conditions. We show that the assumptions of Chernoff’s theorem hold for semi-algebraic sets such that the asymptotics are determined by the tangent cone at the true parameter point. At boundary points or singularities, the tangent cone need not be a linear space and limiting distributions other than chi-square distributions may arise. While boundary points often lead to mixtures of chi-square distributions, singularities give rise to nonstandard limits. We demonstrate that minima of chi-square random variables are important for locally identifiable models, and in a study of the factor analysis model with one factor, we reveal connections to eigenvalues of Wishart matrices.


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Mathias Drton. "Likelihood ratio tests and singularities." Ann. Statist. 37 (2) 979 - 1012, April 2009.


Published: April 2009
First available in Project Euclid: 10 March 2009

zbMATH: 1196.62020
MathSciNet: MR2502658
Digital Object Identifier: 10.1214/07-AOS571

Primary: 60E05 , 62H10

Keywords: Algebraic statistics , factor analysis , large sample asymptotics , semi-algebraic set , tangent cone

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 2 • April 2009
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