Fiducial inference framework for restricted parameter spaces: poisson mean with background

To address the challenge of constructing valid confidence intervals (CIs) for Poisson means in biomedical low-count experiments (e.g., radiation or molecular counting) with known background signals, where existing methods yield overly conservative intervals due to constraints in parameter space. We propose a fiducial framework that redefines the fiducial distribution by adjusting for conditional probability within the restricted parameter space. This computationally efficient approach eliminates empty intervals and leverages parameter constraints to ensure frequentist validity. Numerical simulations demonstrate that the proposed CIs are narrower than conventional methods while maintaining nominal coverage probabilities, particularly near boundary conditions. The method was validated using three real-world biomedical/physics datasets. The fiducial approach provides a robust, statistically efficient solution for Poisson mean inference in restricted spaces. It offers improved precision without compromising coverage, making it highly suitable for analyzing low-count data in biomedical and physical sciences.