On Stability and Decomposition of Sample Quantiles under Heavy-Tailed Distributions

TL;DR

This paper investigates the stability and decomposition of sample quantiles in heavy-tailed distributions, introducing a novel Q-Q orthogonality approach to separate effects of projection direction and quantile threshold estimation.

Contribution

It proposes a new Q-Q orthogonality formulation to distinguish between projection-direction and quantile-threshold effects in heavy-tailed distribution analysis.

Findings

Decomposition of quantile differences into three distinct terms.

Introduction of a Q-Q orthogonality framework for stability analysis.

Enhanced understanding of quantile behavior under heavy tails.

Abstract

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, $\hat q_{\alpha}(\hat w)-q_{\alpha}(w_0)=D_1+D_2+D_3$. 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.