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

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, $\hat q_α(\hat w)-q_α(w_0)=D_1+D_2+D_3$. 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.

• The paper presents a novel Q-Q orthogonality decomposition that splits quantile estimation error into directional, empirical, and remainder components.
• It employs VC-class empirical process theory and symmetric difference bounds to rigorously analyze Value-at-Risk measures in high-dimensional, heavy-tailed settings.
• Monte Carlo experiments validate that weight estimation and empirical quantile fluctuations dominate risk assessment, with the Bahadur remainder remaining a minor yet stable factor.

Stability and Decomposition of Sample Quantiles under Heavy-Tailed Distributions

Introduction

The paper "On Stability and Decomposition of Sample Quantiles under Heavy-Tailed Distributions" (2605.18370) addresses the problem of analyzing sample quantiles—particularly Value-at-Risk (VaR) statistics—in the context where the underlying data is heavy-tailed, and where the quantile is computed not simply for a fixed distribution, but for a distribution indexed by data-driven projection directions. This scenario is central in risk-sensitive financial applications, where returns are linearly projected according to estimated weights, challenging standard asymptotic representations and raising questions regarding the local stability of sample quantiles and their decomposition into interpretable sources of variability.

Problem Formulation and Existing Approaches

In high-dimensional statistical risk management, the focus is typically on the projected portfolio loss $L = -w^\top R$, where $R$ is a random vector of asset returns, and $w$ is a vector of portfolio weights. Both the quantile (e.g., VaR) and the projection direction $w$ are estimated from data. Classical asymptotic analysis for sample quantiles, notably Bahadur's representation, presumes a fixed data distribution, allowing sample quantile fluctuations to be expressed neatly via the empirical CDF and a negligible remainder. However, when $w$ is itself random and estimated, these methods aggregate several sources of instability into a single error term.

Empirical process theory and the geometry of half-spaces provide a global approach: the empirical distribution is controlled uniformly over classes of half-spaces parameterized by $(w, t)$, where $t$ is the quantile threshold. This viewpoint leads to stability bounds in terms of symmetric differences, but these bounds fail to disentangle the individual influences of $w$ (the projection direction) and $t$ (the threshold), and they impose an unnecessarily global uniformity requirement on an intrinsically local problem.

Main Contribution: Q-Q Orthogonality Decomposition

The paper introduces a Q-Q orthogonality framework to achieve a local, interpretable decomposition of sample quantile error under heavy-tailed distributions. Specifically, the error between the empirical quantile at the estimated direction $\hat w$ and the population quantile at a reference direction $R$0 is decomposed as:

$R$1

where:

• $R$2 captures sensitivity of the population quantile to perturbations in projection direction.
• $R$3 is the empirical quantile fluctuation for fixed direction (Bahadur linear term).
• $R$4 is the Bahadur remainder, asymptotically negligible but nontrivial for heavy-tailed models.

This orthogonal decomposition allows the identification and quantification of sources of variance that are obscured by aggregate empirical-process bounds.

Theoretical Results and Asymptotics

The authors establish the following:

• Symmetric Difference Bounds: Under broad conditions, including a local Lipschitz density and finite moments, the probability mass of the symmetric difference between two half-spaces—one indexed by $R$5, one by $R$6—is controlled via the size of the perturbation in weights and threshold.
• Multivariate $R$7-Model Specialization: When $R$8 with $R$9, all directional projections possess smooth, bounded densities with explicit control. The symmetric difference bounds become analytically explicit in terms of $w$0, $w$1, and $w$2.
• Empirical Process Control: By leveraging the VC-class structure of half-space indicators, uniform Glivenko–Cantelli convergence applies, allowing empirical probabilities to uniformly approximate population probabilities even under indexed perturbations.
• Q-Q Orthogonality Theorem: Under regularity (empirical-process uniformity, local density regularity, and $w$3-consistent weight estimation), the decomposition above holds with $w$4, $w$5, and $w$6. The result immediately yields consistency of the estimated quantile:

$w$7

Interpretation of the Q-Q Orthogonality Decomposition

• $w$8, Directional Perturbation: This term quantifies the population-level instability in VaR solely due to uncertainty in the data-driven weight vector. It is highly interpretable and essential for understanding and mitigating risk in high-dimensional finance.
• $w$9, Empirical Quantile Fluctuation: This follows classic quantile CLT. For fixed $w$0, sample quantile variability is well-understood; by indexing at the random $w$1, the analysis generalizes this to data-driven directions.
• $w$2, Bahadur Remainder: Even with heavy-tailed data ($w$3 close to $w$4), the nominal $w$5 rate persists. However, the constants grow as tail thickness increases, reflecting practical instability in VaR estimation in extreme regimes.

The decomposition brings clarity to the relative importance of weight estimation errors versus quantile estimation errors, and makes explicit the limitations of aggregate symmetric-difference-based bounds.

Numerical Evidence

Through Monte Carlo experiments with projected multivariate $w$6 distributions, the authors quantify all summands in the decomposition for various $w$7 and quantile levels $w$8 (including extreme values). Empirical findings include:

• $w$9 and $w$0 dominate estimation error, with $w$1 contributing a stable, low percentage across a range of regimes (4-8\%).
• The Bahadur remainder $w$2 is strongly affected in magnitude, but not asymptotic rate, by tail thickness and proximity to the quantile tails.
• Fitted slopes for $w$3 versus $w$4 are consistent with the $w$5 rate, even for $w$6.

Implications and Future Directions

The decomposition has significant implications for both theory and practice:

• It enables robust uncertainty quantification for risk measures under heavy tails, attributing uncertainty to estimation of weights versus intrinsic sampling variability.
• The results generalize classical quantile asymptotics and deliver finite-sample guidance for stress regimes, a necessity for financial applications under non-Gaussianity.
• The analytical separation provides a natural avenue for Bayesian inference on VaR via the posterior predictive distribution of $w$7 as $w$8 varies.
• Extensions to dependent data, non-linear projections, and more general classes of functionals are indicated as fertile ground for further research.

Conclusion

This work provides a rigorous, modular analysis of the sources of sample quantile instability in high-dimensional, heavy-tailed settings. The Q-Q orthogonality decomposition clarifies the impact of projection direction error and quantile empirical fluctuation, with strong asymptotic and finite-sample justification. The approach advances the understanding of statistical risk measures in non-ideal settings, supporting robust inference and mitigation strategies for heavy-tailed risk, and lays the foundation for subsequent theoretical generalizations and practical improvements in the field.