A Monte Carlo analysis was conducted to explore the effects of a constraint boundary on the true size of the confidence limits computed by two methods, (1) inversion of the Wald statistic, and (2) inversion of likelihood ratio statistic.
The 95 percent likelihood ratio and Wald confidence limits for means constrained to be greater than zero were estimated for 40 models: a Normal with unit variance and 20 different true values for the means ranging from 0 to .18 for each of two sample sizes, 300 and 500.
The proportion of the confidence limits that failed to include the true value
is plotted in Figure 1 against the true value. We observe that this
proportion, or size, is about one half the correct size up to a
threshold where it becomes the full correct size. The theoretical
thresholds for confidence limits for a mean with Normal density
with N = 300 and 500, at 
,
are .1131, .0876, and for 
are .0950, .0736,
respectively. These threshold values are
quite close to the Monte Carlo results in Figure 1.
Figure 1: Size of constrained parameter of interest at different
distances from the constraint boundary