Constrained Parameters in Applications: Review of Issues and Approaches

Table 1

Upper percentiles of the asymptotic distribution of the LRT in case of one parameter of interest and two nuisance parameters on the boundary, calculated by simulations. Correlations are between parameter of interest and nuisance parameter 1, parameter of interest and nuisance parameter 2, and nuisance parameters 1 and 2. The last row shows corresponding percentiles of the chi-square with one degree of freedom—the asymptotic distribution of the LRT when complications due to boundary parameters are ignored. Percentiles exceeding corresponding percentiles of 𝜒² (1), that is, situations when use of 𝜒² (1) is anticonservative, are bolded.

Correlation between model parameters	Percentiles
	95th	99th
0.8, 0.8, 0.6	4.690	7.933
0.2, 0.8, −0.2	4.409	7.569
0.2, 0.8, 0.3	3.808	6.802
0.2, 0.8, 0.6	3.413	6.270
0.5, 0.5, −0.2	4.317	7.476
0.5, 0.5, 0.3	3.910	6.983
0.5, 0.5, 0.6	3.729	6.786
0.2, 0.2, −0.8	3.756	6.802
0.2, 0.2, 0.9	3.021	5.683
0, 0.8, −0.5	4.440	7.575
0, 0.8, 0.5	3.060	5.762
−0.5, −0.5, 0.6	1.607	3.659
−0.8, 0.8, 0.8	0.848	2.188
𝜒^2 (1)	3.841	6.835