Likelihood ratio tests and singularities
Mathias Drton
Source: Ann. Statist. Volume 37, Number 2 (2009), 979-1012.
Abstract
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. If time permits we will
also briefly discuss related issues for Wald tests and the Bayesian
information criterion.