  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.
