 Location-scale invariant simulation test for a Gaussian variance component

With fixed $X,Z$, compare $N(X\beta ,{\sigma }^{2}I)$ and $N(X\beta ,{\sigma }^{2}I+{\tau }^{2}Z{Z}^T)$ by maximizing the ordinary likelihood function in both models. Their likelihood-ratio test statistic is unchanged by $Y\mapsto Xb+cY$, $c>0$, because both maximized log-likelihoods change by the same $-n\log c$. Its null law can therefore be simulated using independent $N(0,I)$ responses. Comparing the observed statistic with simulated statistics by $(1+\#\{T_b\ge T_{obs}\})/(B+1)$ gives a conservative finite-sample Monte Carlo p-value, subject to correctly maximizing both likelihoods.