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.

翻译：我们研究由估计参数索引的分布下的样本分位数，重点关注与金融收益线性投影相关的风险价值，这些收益的基础概率分布具有重尾性。在此设定中，投影方向和经验分位数阈值均从数据中估计得出，因此固定分布下的标准Bahadur表示无法分离不同的不稳定性来源。一个规范的出发点是Bahadur表示，它通过经验分布函数加上余项来表示样本分位数 \cite{bahadur1966}。经验过程理论通过半空间、对称差以及Glivenko--Cantelli一致收敛的机制提供了可用的框架。这些方法给出了稳定性界，但将投影方向的变化和分位数阈值的变化吸收到单个对称差度量中。有趣的是，这本质上是一个局部分位数稳定性问题，却施加了全局一致收敛的要求。本文引入了一种Q-Q正交性表述，用于分离投影方向和分位数阈值的影响。我们关注的对象是使用估计投影方向计算的经验分位数与参考投影方向下的总体分位数之间的差异。我们将此差异分解为三项：$\hat q_α(\hat w)-q_α(w_0)=D_1+D_2+D_3$。其中，$D_1$ 衡量由投影方向扰动引起的总体分位数移动，$D_2$ 衡量在投影方向固定不变时经验分位数的波动，$D_3$ 是Bahadur型余项。