We establish asymptotic results for profile score statistics under sequences of parameters that may tend to the boundary of the parameter set. When the likelihood is profiled over a set with boundary points, the distribution of the statistic can depend, even asymptotically, on a distance to the boundary that cannot be estimated consistently. We address this in three ways. First, we give conditions under which the nuisance score has a root on an extended parameter set, and the profile score statistic formed with that root converges to a chi-squared distribution. Second, we show that the efficient score statistic converges to a chi-squared distribution under a separate condition that can hold even for estimators constrained to the original set. Both lead to confidence regions with asymptotically correct coverage probability, uniformly on sets that may contain boundary points, and the conditions allow the scores for the interest and nuisance parameters to grow at different rates or become asymptotically collinear. Third, we quantify the coverage error of applying the profile score statistic without an extended set. We approximate the distribution of the statistic and bound the error by a quantity that decays like a normal tail in the standardized distance to the boundary. The approximation also gives one-sided tests with least favorable critical values whose asymptotic size is at most the nominal level. We illustrate the results in a zero-inflated Poisson model and a one-way random effects model.

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