A note on the asymptotic distribution of the Likelihood Ratio Test statistic under boundary conditions

In the context of likelihood ratio testing with parameters on the boundary, we revisit two situations for which there are some discrepancies in the literature: the case of two parameters of interest on the boundary, with all other parameters in the interior, and the case where one of the two parameters on the boundary is a nuisance parameter. For the former case, we clarify that two seemingly conflicting results are consistent upon closer examination. For the latter, we clarify the source of the discrepancy and explain the different findings. As for this case the closed-form expression is valid only under positive correlation, we further propose a heuristic modification to the asymptotic distribution of the likelihood ratio test that extends its applicability to cases involving negative correlation.