分离投影方向与分位数阈值对风险估计的影响，提升重尾分布下风险预测的稳定性。
On Stability and Decomposition of Sample Quantiles under Heavy-Tailed Distributions
- 提出Q-Q正交分解框架，解耦方向与阈值变化对分位数的影响
- 证明三重分解中各分量可分别控制，实现全局稳定边界
- 适用于金融风险中的价值风险（VaR）计算，适合量化研究者
研究在参数估计依赖的重尾分布下样本分位数的稳定性问题，聚焦于金融收益线性投影相关的风险价值（Value-at-Risk）。由于投影方向和经验分位数阈值均从数据中估计，标准的Bahadur表示无法区分两类扰动来源。本文基于半空间、对称差与Glivenko-Cantelli一致收敛构建理论框架，但传统方法将方向与阈值变化合并为单一对称差度量。为此，提出一种新的Q-Q正交性表述，将估计方向下的经验分位数与参考方向下总体分位数之差分解为三项：D₁衡量方向扰动引起的总体分位数移动；D₂衡量固定方向下经验分位数波动；D₃为类Bahadur余项。该分解实现了对关键影响源的独立分析，提升了重尾环境下风险估计的可靠性。
原文摘要 · Abstract (English)
We study sample quantiles of distributions indexed by estimated parameters, with a on Value-at-Risk related to linear projections of financial returns that whose underlying probability law is heavy-tailed. In this setting, the projection direction and the empirical quantile threshold are estimated from the data, so the standard Bahadur representation under a fixed distribution does not separate the distinct sources of instability. A canonical starting point is Bahadur's representation, which expresses the sample quantile through the empirical distribution function plus a remainder term \cite{bahadur1966}. Empirical-process theory provides a usable scaffolding through the mechanics of half-spaces, symmetric differences, and Glivenko--Cantelli uniform convergence. They yield stability bounds, but absorb changes in projection direction and changes in quantile threshold into a single symmetric-difference measure. Interestingly, a global uniform-convergence requirement is imposed on what is intrinsically a local quantile-stability problem. This paper introduces a Q-Q orthogonality formulation for separating projection-direction and quantile-threshold effects. The object of interest is the difference between the empirical quantile computed using the estimated projection direction and the population quantile computed at the reference projection direction. We decompose this difference into three terms, $\hat q_α(\hat w)-q_α(w_0)=D_1+D_2+D_3$. Here, $D_1$ measures the population quantile movement induced by perturbing the projection direction, $D_2$ measures the empirical quantile fluctuation with the projection direction held fixed, and $D_3$ is the Bahadur-type remainder.
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