The stratified linear permutation statistic arises in various statistics problems, including stratified and post-stratified survey sampling, stratified and post-stratified experiments, conditional permutation tests, etc. Although we can derive the Berry--Esseen bounds for the stratified linear permutation statistic based on existing bounds for the non-stratified statistics, those bounds are not sharp, and moreover, this strategy does not work in general settings with heterogeneous strata with varying sizes. We first use Stein's method to obtain a unified stratified permutational Berry--Esseen bound that can accommodate heterogeneous strata. We then apply the bound to various statistics problems, leading to stronger theoretical quantifications and thereby facilitating statistical inference in those problems.

翻译：分层线性置换统计量出现在多种统计学问题中，包括分层与后分层调查抽样、分层与后分层实验、条件置换检验等。尽管我们可以基于非分层统计量的现有界推导出分层线性置换统计量的Berry--Esseen界，但这些界并不尖锐，且该策略在具有不同规模异质层的通用设置中通常失效。我们首先利用Stein方法获得一个统一的分层置换Berry--Esseen界，该界能够适应异质层的情况。随后，我们将此界应用于多种统计学问题，从而得到更强的理论量化结果，并促进这些问题的统计推断。