中位数与HulC推广的推断
Inference for Median and a Generalization of HulC
AI总结：
本文提出HulC推广，对渐近正态估计量比Wald区间窄，且对非正态极限分布问题也有效。通过分析中位数无分布置信区间宽度特性，发现适当缩放后区间宽度收敛到非退化随机变量。
摘要：
文献记载样本分割在统计推断中具有显著的方法和理论优势，如双机器学习中的交叉拟合、参数推断的通用推断和分裂共形预测等，新提出的HulC方法也属此类。HulC将目标函数视为估计量的近似中位数并应用经典中位数无分布置信区间。当估计量渐近正态时，HulC区间比Wald区间平均宽50%。本文提出HulC推广，对渐近正态估计量仅比Wald区间宽25%，且对非正态极限分布的更广泛问题有效。同时分析了中位数处勒贝格密度为零或无穷时中位数的无分布置信区间宽度特性，发现适当缩放后区间宽度收敛到非退化随机变量。
英文摘要：
It is well-documented in the literature that sample splitting offers significant methodological and theoretical advantages in statistical inference. This, for example, includes cross-fitting in double machine learning, universal inference for parametric inference, and split conformal prediction. The recently proposed inference method, HulC, also falls into this category. HulC operates by viewing the target functional of interest as an approximate median of an estimator and applying the classical distribution-free confidence intervals for median with minimal sample size. When the estimators are asymptotically normal, HulC intervals are shown to be 50\% wider than the Wald intervals asymptotically, on average, for $95\%$ coverage. Interestingly, this ratio of widths converges to a non-degenerate distribution. In this paper, we propose a generalization of HulC that are only 25\% wider than the Wald intervals for asymptotically normal estimators, irrespective of the nominal coverage. Furthermore, similarly to HulC, these generalized intervals remain valid for a significantly wider class of problems with non-normal limiting distributions. To better understand width properties under non-normal limiting distributions, we analyze distribution-free confidence intervals for the median when the Lebesgue density at the median is either zero or infinite. Surprisingly, we find that properly scaled, the interval width converges to a non-degenerate random variable.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
