Insights into Tail-Based and Order Statistics

ArXiv ID: 2511.04784 “View on arXiv”

Authors: Hamidreza Maleki Almani

Abstract

Heavy-tailed phenomena appear across diverse domains –from wealth and firm sizes in economics to network traffic, biological systems, and physical processes– characterized by the disproportionate influence of extreme values. These distributions challenge classical statistical models, as their tails decay too slowly for conventional approximations to hold. Among their key descriptive measures are quantile contributions, which quantify the proportion of a total quantity (such as income, energy, or risk) attributed to observations above a given quantile threshold. This paper presents a theoretical study of the quantile contribution statistic and its relationship with order statistics. We derive a closed-form expression for the joint cumulative distribution function (CDF) of order statistics and, based on it, obtain an explicit CDF for quantile contributions applicable to small samples. We then investigate the asymptotic behavior of these contributions as the sample size increases, establishing the asymptotic normality of the numerator and characterizing the limiting distribution of the quantile contribution. Finally, simulation studies illustrate the convergence properties and empirical accuracy of the theoretical results, providing a foundation for applying quantile contributions in the analysis of heavy-tailed data.

Keywords: Heavy tails, Quantile contributions, Order statistics, Asymptotic normality, Risk measures, General Financial Data

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is heavily theoretical, deriving closed-form CDFs for order statistics and asymptotic distributions with advanced probability theory, which drives a high math score. However, the empirical component is limited to simulation studies illustrating convergence properties, lacking backtesting or real-world data implementation, resulting in low empirical rigor.

  flowchart TD
    A["Research Goal:<br>Quantify Heavy-Tail Influence<br>via Quantile Contributions"] --> B["Key Methodology<br>Theoretical Derivation & Simulation"]
    B --> C["Data/Inputs<br>Heavy-Tailed Distributions<br>Small & Large Samples"]
    C --> D["Computational Process 1:<br>Derive Joint CDF of Order Statistics"]
    D --> E["Computational Process 2:<br>Calculate Exact CDF<br>for Quantile Contributions"]
    E --> F["Computational Process 3:<br>Simulate Asymptotic Behavior<br>& Verify Normality"]
    F --> G["Key Findings<br>Closed-form CDFs, Asymptotic Normality,<br>Empirical Validation for Risk Analysis"]