Likelihood ratio tests and singularities

Details Likelihood ratio tests and singularities Likelihood ratio tests and singularities

2007-03-12

arxiv.org/abs/math/0703360v3

Annals of Statistics 2009, Vol. 37, No. 2, 979-1012

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/07-AOS571 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

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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