 Variance-component likelihood-ratio test at a boundary

Testing a single nonnegative random-effect variance against zero puts the null on the boundary of the parameter space. In the usual regular increasing-independent-groups limit, the likelihood-ratio test statistic has limit $\frac{1}{2}{\delta }_0+\frac{1}{2}{\chi }_1^2$, rather than ${\chi }_1^2$. A finite-sample simulation calibrated for the actual design avoids relying on this asymptotic approximation.