Near-boundary asymptotics for profile score statistics
Abstract
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.
Related
- Exact Asymptotics for the Harmonic Descent Chain and Extended Convergence Bounds for Recursive Statistics
- Uniform Asymptotics for Andrews-Beck Partition Statistics
- Statistics on Yau's conjecture: Variance asymptotics
- Boundary-layer asymptotics for Gaussian-smoothed singular measures
- On the Large Λ Asymptotics of the One Phase Bernoulli Free Boundary Problem
BibTeX
@misc{shedden2026nearboundary,
title = {{Near-boundary asymptotics for profile score statistics}},
author = {Matias Shedden and Karl Oskar Ekvall},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33912},
eprint = {2610.33912},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33912},
}
RIS
TY - UNPB
TI - Near-boundary asymptotics for profile score statistics
AU - Matias Shedden
AU - Karl Oskar Ekvall
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33912
UR - https://arxiv.org/abs/2610.33912
AB - 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.
ER -