Confidence intervals for a Poisson parameter with background

Due to the discrete nature and parameter constraints, the interval estimation of a Poisson parameter with background has been a challenging problem in statistics. Among the existing good methods, the FC interval, RW interval, G interval, MS interval, and EB interval have conservative coverage probability in our simulation study. In this paper, we propose a randomized confidence interval (CI) for a constrained Poisson parameter based on the inferential model (IM) and suggest the practical use of its nonrandomized approximation. The randomized IM CI is proven to guarantee the exact coverage probability, and our nonrandomized approximation has coverage closer to the confidence coefficient than the existing CIs in most cases. Moreover, our nonrandomized interval always has the shortest expected length among the six CIs. Finally, a real example is used to demonstrate the application of the proposed methods.