We propose a density-free method for frequentist inference on population quantiles, termed Self-Normalized Quantile Empirical Saddlepoint Approximation (SNQESA). The approach builds a self-normalized pivot from the indicator score for a fixed quantile threshold and then employs a constrained empirical saddlepoint approximation to obtain highly accurate tail probabilities. Inverting these tail areas yields confidence intervals and tests without estimating the unknown density at the target quantile, thereby eliminating bandwidth selection and the boundary issues that affect kernel-based Wald/Hall-Sheather intervals. Under mild local regularity, the resulting procedures attain higher-order tail accuracy and second-order coverage after inversion. Because the pivot is anchored in a bounded Bernoulli reduction, the method remains reliable for skewed and heavy-tailed distributions and for extreme quantiles. Extensive Monte Carlo experiments across light, heavy, and multimodal distributions demonstrate that SNQESA delivers stable coverage and competitive interval lengths in small to moderate samples while being orders of magnitude faster than large-B resampling schemes. An empirical study on Value-at-Risk with rolling windows further highlights the gains in tail performance and computational efficiency. The framework naturally extends to two-sample quantile differences and to regression-type settings, offering a practical, analytically transparent alternative to kernel, bootstrap, and empirical-likelihood methods for distribution-free quantile inference.

翻译：我们提出了一种用于频率推断中总体分位数的无密度方法，称为自归一化分位数经验鞍点逼近（SNQESA）。该方法基于固定分位数阈值的指示得分构建自归一化枢轴量，然后采用约束经验鞍点逼近来获得高精度的尾部概率。通过反转这些尾部区域，可在无需估计目标分位数处未知密度的情况下获得置信区间和检验，从而避免了带宽选择以及影响基于核的Wald/Hall-Sheather区间的边界问题。在温和的局部正则性条件下，所得方法在反转后实现了高阶尾部精度和二阶覆盖概率。由于枢轴量基于有界伯努利简化构建，该方法对于偏态分布、重尾分布以及极端分位数仍保持可靠性。在轻尾、重尾和多峰分布上进行的大量蒙特卡洛实验表明，SNQESA在小到中等样本量下能提供稳定的覆盖率和具有竞争力的区间长度，同时计算速度比大B重采样方案快数个数量级。一项基于滚动窗口的风险价值实证研究进一步凸显了该方法在尾部性能和计算效率方面的优势。该框架可自然扩展到两样本分位数差异及回归类型设定，为无分布分位数推断提供了一种实用、解析透明的替代方案，优于核方法、自助法和经验似然方法。