  The likelihood ratio statistic, with its asymptotic $\chi^2$ distribution at
regular model points, is often used for hypothesis testing. At model
singularities and boundaries, however, the asymptotic distribution may not be
$\chi^2$, as highlighted by recent work of Drton. Indeed, poor behavior of a
$\chi^2$ 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 $\chi^2$.
