Estimation frameworks for statistical inference are preferred to hypothesis testing when quantifying uncertainty and precise estimation are more valuable than binary decisions about statistical significance. Study design for estimation-based investigations often uses precision criteria to select sample sizes that control the length of interval estimates with respect to a sampling distribution. In this paper, we formally define a distribution that characterizes the probability of obtaining a sufficiently narrow interval estimate as a function of the sample size. This distribution can be used to determine the smallest sample size needed to ensure an interval estimate is sufficiently narrow. We prove that this distribution is approximately normal in large-sample settings for many data generation processes. However, this approximate normality may not hold for studies with moderate sample sizes, particularly when incorporating prior information or obtaining asymmetric interval estimates. Thus, we also propose an efficient simulation-based approach to approximate the distribution for the sample size by estimating the sampling distribution of interval estimate lengths at only two sample sizes. Our methodology provides a unified framework for design with precision criteria in Bayesian and frequentist settings. We illustrate the broad applicability of this framework with several examples.

翻译：在量化不确定性和精确估计比统计显著性的二元决策更有价值时，统计推断的估计框架通常优于假设检验。基于估计的研究设计常采用精度准则来选择样本量，以控制抽样分布下区间估计的长度。本文正式定义了一种分布，该分布将获得足够窄区间估计的概率表征为样本量的函数。此分布可用于确定确保区间估计足够窄所需的最小样本量。我们证明，对于许多数据生成过程，该分布在大样本场景下近似服从正态分布。然而，对于中等样本量的研究，尤其是在纳入先验信息或获得非对称区间估计时，这种近似正态性可能不成立。因此，我们还提出一种高效的基于模拟的方法，通过仅在两个样本量下估计区间估计长度的抽样分布来近似样本量的分布。我们的方法为贝叶斯和频率主义框架下基于精度准则的设计提供了统一框架，并通过多个实例展示了该框架的广泛适用性。