Constructing distribution-free confidence intervals for the median, a classic problem in statistics, has seen numerous solutions in the literature. While coverage validity has received ample attention, less has been explored about interval width. Our study breaks new ground by investigating the width of these intervals under non-standard assumptions. Surprisingly, we find that properly scaled, the interval width converges to a non-degenerate random variable, unlike traditional intervals. We also adapt our findings for constructing improved confidence intervals for general parameters, enhancing the existing HulC procedure. These advances provide practitioners with more robust tools for data analysis, reducing the need for strict distributional assumptions.

翻译：构建中位数的无分布置信区间是统计学中的经典问题，已有诸多文献提出相应解决方案。尽管区间覆盖的可靠性已得到充分研究，但其区间宽度问题却鲜有探讨。本研究在非标准假设下创新性地考察了此类区间的宽度性质。令人意外的是，我们发现经适当缩放后，区间宽度收敛于一个非退化随机变量——这与传统区间截然不同。此外，我们将研究成果推广至一般参数的改进置信区间构建，优化了现有HulC方法。这些进展为数据分析实践提供了更稳健的工具，降低了对严格分布假设的依赖。