Hypothesis testing near singularities and boundaries

Abstract

The likelihood ratio statistic, with its asymptotic $\chi ^{2}$ distribution at regular model points, is often used for hypothesis testing. However, the asymptotic distribution can differ at model singularities and boundaries, suggesting the use of a $\chi ^{2}$ might be problematic nearby. Indeed, its poor behavior for testing near singularities and boundaries is apparent in simulations, and can lead to conservative or anti-conservative tests. Here we develop a new distribution designed for use in hypothesis testing near singularities and boundaries, which asymptotically agrees with that of the likelihood ratio statistic. For two example trinomial models, arising in the context of inference of evolutionary trees, we show the new distributions outperform a $\chi ^{2}$.