Constructing valid inferential methods for constrained parameters in normal and Poisson distributions represents two fundamental and important problems in applied statistics, for which there is currently no unified framework for statistical inference. Most existing studies assume that the nuisance parameters of the model are known, an assumption that is often impractical in real-world applications. However, under the more realistic scenario where nuisance parameters are unknown, the available Bayesian interval estimation methods fail to guarantee nominal coverage and thus cannot provide exact inference. To address these limitations, this paper develops prior-free inferential model (IM) approaches for parameters of interest in constrained normal and Poisson models and demonstrates that the confidence intervals (CIs) obtained from these novel IM methods can achieve exact nominal coverage. Furthermore, considering the discrete nature of the Poisson distribution, we employ random weighting techniques to improve the conservative coverage performance of the IM CIs. Simulation studies show that the coverage probabilities of the improved nonrandomized inferential model (NIM) CIs are closest to the prespecified nominal levels, with corresponding expected lengths shorter than those of Bayesian intervals in weak signal scenarios, whereas the shorter expected lengths of Bayesian intervals in strong signal scenarios come at the cost of sacrificing coverage guarantees. Therefore, the proposed IM and NIM CIs are superior to the Bayesian CIs. Finally, the advantages of the proposed methods are confirmed through an analysis of two experimental datasets on neutrinos in high-energy physics.

翻译：为服从正态分布与泊松分布的约束参数构建有效的推断方法，是应用统计学中两项基础且重要的问题，目前尚无统一的统计推断框架。现有研究大多假设模型的 nuisance 参数已知，这一假设在现实应用中往往难以成立。然而，在更符合实际的 nuisance 参数未知场景下，现有的贝叶斯区间估计方法无法保证名义覆盖水平，从而无法提供精确推断。为克服这些局限，本文针对约束正态模型与约束泊松模型中的目标参数，提出了无先验推断模型方法，并证明由这些新型推断模型方法获得的置信区间能够实现精确的名义覆盖水平。此外，鉴于泊松分布的离散特性，我们采用随机加权技术以改善推断模型置信区间的保守覆盖性能。模拟研究表明，改进后的非随机化推断模型置信区间的覆盖概率最接近预设的名义水平，在弱信号场景下其对应期望长度短于贝叶斯区间，而强信号场景下贝叶斯区间的期望长度更短，但其代价是牺牲了覆盖保证。因此，本文提出的推断模型与非随机化推断模型置信区间优于贝叶斯置信区间。最后，通过对高能物理中两组中微子实验数据集的分析，进一步证实了所提方法的优势。