Likelihood ratio tests and singularities. (English) Zbl 1196.62020

Summary: 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.


62F05 Asymptotic properties of parametric tests
62H25 Factor analysis and principal components; correspondence analysis
62H15 Hypothesis testing in multivariate analysis
62H10 Multivariate distribution of statistics
62E20 Asymptotic distribution theory in statistics


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