Likelihood Ratio Tests and Singularities
Monday, March 5, 2007 - 10:40am - 11:30am
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 such that non-standard limiting distributions may arise. Besides the well-known mixtures of chi-square distributions, such non-standard limits are shown to include the distributions of minima of chi-square random variables. Via algebraic tangent cones, connections to eigenvalues of Wishart matrices are found in factor analysis.