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

2026-05-19 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Research Methodology & Innovation, Capital Markets & Investment Management · Depth: Expert, extended

Summary

This paper addresses the instability of sample quantiles, particularly Value-at-Risk (VaR), when estimated parameters like projection direction and quantile threshold are derived from heavy-tailed financial returns. It introduces a novel Q-Q orthogonality formulation that decomposes the total difference between the empirical and reference population quantiles into three distinct components: $D_1$ measures population quantile movement due to changes in the projection direction, $D_2$ captures empirical quantile fluctuation with the direction fixed, and $D_3$ is the Bahadur-type remainder. The framework leverages empirical process theory and multivariate Student $t_{\nu}$ distributions, demonstrating that while heavy tails inflate the absolute magnitude of these components, the classical Bahadur-Kiefer $n^{-3/4}$ remainder rate persists. Numerical experiments validate this decomposition, showing that $D_1$ and $D_2$ are the dominant error sources, and the remainder's decay rate remains stable across varying tail regimes. This decomposition provides a clearer understanding of VaR uncertainty by separating directional perturbation, empirical fluctuation, and higher-order residual effects.

Key takeaway

A novel Q-Q orthogonality decomposition separates the total error in estimated sample quantiles (e.g., VaR) under heavy-tailed distributions and estimated projection directions into three components: population directional shifts ($D_1$), empirical fluctuations ($D_2$), and a Bahadur remainder ($D_3$). Validated with multivariate Student $t_{\nu}$ distributions ($\nu>2$), $D_1$ and $D_2$ account for the majority of error, while $D_3$ maintains its classical $O_P(n^{-3/4}\log n)$ rate, though its magnitude increases with heavier tails and extreme quantiles. This framework provides a robust method for stability analysis and inference in quantitative finance, offering a clearer understanding of uncertainty sources by disentangling estimation errors from sampling variability.

Topics

• Sample Quantiles
• Heavy-Tailed Distributions
• Value-at-Risk
• Q-Q Orthogonality Decomposition
• Bahadur Representation

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.