Likelihood-Based Confidence Curves for Bounded Parameters
Abstract
When a parameter is bounded, the likelihood ratio statistic does not generally follow the chi-square limit distribution of Wilks' theorem. In fact, its asymptotic null distribution depends on the distance from the null value to the boundary, interpolating between the chi-bar-square law on the boundary and the chi-square law in the deep interior of the parameter space. We derive this distribution and use it to construct confidence curves, which are nested collections of two-sided confidence intervals or regions, for bounded parameters. We extend the existing result for the scalar parameter to a two-dimensional parameter with one or both components bounded. As the bivariate distributions are integrals of normal distributions, which are not reducible to standard statistical functions, we also give a simpler approximated distribution obtained by moment-matching. We illustrate the methods on a canonical analysis from high-energy physics, namely the Poisson signal-plus-background experiment.
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BibTeX
@misc{reiersrud2026likelihoodbased,
title = {{Likelihood-Based Confidence Curves for Bounded Parameters}},
author = {Hedvig Borgen Reiersrud and Clara Bertinelli Salucci and Riccardo De Bin and Alessandra Rosalba Brazzale},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28235},
eprint = {2610.28235},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28235},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Likelihood-Based Confidence Curves for Bounded Parameters
AU - Hedvig Borgen Reiersrud
AU - Clara Bertinelli Salucci
AU - Riccardo De Bin
AU - Alessandra Rosalba Brazzale
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28235
UR - https://arxiv.org/abs/2610.28235
AB - When a parameter is bounded, the likelihood ratio statistic does not generally follow the chi-square limit distribution of Wilks' theorem. In fact, its asymptotic null distribution depends on the distance from the null value to the boundary, interpolating between the chi-bar-square law on the boundary and the chi-square law in the deep interior of the parameter space. We derive this distribution and use it to construct confidence curves, which are nested collections of two-sided confidence intervals or regions, for bounded parameters. We extend the existing result for the scalar parameter to a two-dimensional parameter with one or both components bounded. As the bivariate distributions are integrals of normal distributions, which are not reducible to standard statistical functions, we also give a simpler approximated distribution obtained by moment-matching. We illustrate the methods on a canonical analysis from high-energy physics, namely the Poisson signal-plus-background experiment.
ER -