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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. [ABSTRACT FROM AUTHOR]

Interval estimation with accurate coverage for risk difference (RD) in a correlated 2 × 2 table with structural zero is a fundamental and important problem in biostatistics. The score test-based and Bayesian tail-based confidence intervals (CIs) have good coverage performance among the existing methods. However, as approximation approaches, they have coverage probabilities lower than the nominal confidence level for finite and moderate sample sizes. In this paper, we propose three new CIs for RD based on the fiducial, inferential model (IM) and modified IM (MIM) methods. The IM interval is proven to be valid. Moreover, simulation studies show that the CIs of fiducial and MIM methods can guarantee the preset coverage rate even for small sample sizes. More importantly, in terms of coverage probability and expected length, the MIM interval outperforms other intervals. Finally, a real example illustrates the application of the proposed methods. [ABSTRACT FROM AUTHOR]