Prior-free probabilistic interval estimation for constrained parameters

In applied statistics, constructing exact confidence intervals for constrained normal means and constrained Poisson means are two long-standing challenges. Most existing inferential methods assume that the nuisance parameters in constrained models are known constants, which is often impractical. When nuisance parameters are unknown, Bayesian intervals fail to guarantee nominal coverage. To address these issues, this work developed a valid prior-free inferential model approach for normal and Poisson distributions with unknown nuisance parameters. For the constrained normal case, we constructed an inferential model interval that exactly attains the prespecified coverage probability. For the constrained Poisson case, we first proposed an inferential model interval that achieves exact frequentist coverage control; however, owing to the discreteness of the Poisson distribution, this interval is conservative. We then improved it by introducing random weighting, yielding a nonrandomized inferential model method. Simulation studies showed that the inferential model interval achieves exact coverage under the constrained normal model, but is conservative in constrained Poisson inference. In contrast, the nonrandomized inferential model interval attains coverage closest to the nominal level by shortening the interval length. Finally, two neutrino datasets from high-energy physics were used to demonstrate the advantages of the proposed new methods over Bayesian approaches.