Paper: arXiv 2511.04784

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.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: 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.

Research Flowchart

  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"]

Cited References (59)

  1. Estimating the overdispersion in COVID-19 transmission using outbreak sizes outside China. (2020) — link
  2. By Hand (2019) — link
  3. Fat-Tailed Distributions: Data, Diagnostics and Dependence (2014)
  4. On the Super-Additivity and Estimation Biases of Quantile Contributions (2014) — link
  5. The log-dynamic brain: how skewed distributions affect network operations (2014) — link
  6. The 1% Rule in Four Digital Health Social Networks: An Observational Study (2014) — link
  7. Managing the unmanageable: rules, tools, and insights for managing software people and teams by Micky W. Mantle and Ron Lichty (2012) — link
  8. An Introduction to Heavy-Tailed and Subexponential Distributions (2011) — link
and 51 more
  1. The Black Swan: The Impact of the Highly Improbable (2008) — link
  2. The (Mis)Behaviour of Markets: A Fractual View of Risk, Ruin and Reward (2008) — link
  3. Stochastic analysis of file-swarming systems (2007) — link
  4. Power-Law Distributions in Empirical Data (2007) — link
  5. Density of the Ratio of Two Normal Random Variables and Applications (2006) — link
  6. Superspreading and the effect of individual variation on disease emergence (2005) — link
  7. Why Stock Markets Crash: Critical Events in Complex Financial Systems (2005)
  8. INAUGURAL ARTICLE by a Recently Elected Academy Member:Scaling theory for quasibrittle structural failure (2004) — link
  9. Gender and the 10/90 gap in health research. (2004) — link
  10. Network biology: understanding the cell’s functional organization (2004) — link
  11. Size-based scheduling to improve web performance (2003) — link
  12. A signal analysis of network traffic anomalies (2002) — link
  13. The structural cause of file size distributions (2001) — link
  14. Lethality and centrality in protein networks (2001) — link
  15. Empirical properties of asset returns: stylized facts and statistical issues (2001) — link
  16. Non-Gaussian Noise Models in Signal Processing for Telecommunications: New Methods and Results for Class A and Class B Noise Models (1999) — link
  17. Scaling and criticality in a stochastic multi-agent model of a financial market (1999) — link
  18. Self-similarity in World Wide Web traffic: evidence and possible causes (1996) — link
  19. Wide-area traffic: the failure of Poisson modeling (1994) — link
  20. A First Course in Order Statistics (1993) — link
  21. Approximate Distributions of Order Statistics: With Applications to Nonparametric Statistics (1989) — link
  22. Introduction to probability and mathematical statistics (1986) — link
  23. Programmimg pearls (1985) — link
  24. OBTAINING CONFIDENCE INTERVALS FOR THE RISK RATIO IN COHORT STUDIES (1978) — link
  25. On the ratio of two correlated normal random variables (1969) — link
  26. A Note on Quantiles in Large Samples (1966) — link
  27. THE DISTRIBUTION OF THE INDEX IN A NORMAL BIVARIATE POPULATION (1932) — link
  28. 10/90 gap , vol. from the original on January 21, 2021 (2011)
  29. Power Laws in Economics and Finance∗ (2008)
  30. Global notes: the 10/90 gap disparities in global health research (2006) — link
  31. The 10/90 report on health research 2003-2004. (2004)
  32. 10/90 report on health research 2003-2004 , vol. Archived 2015-04-16 at the Wayback Machine (2004)
  33. Steady-state properties of GI/G/1 (2003)
  34. Stochastic processes in insurance and finance (2001) — link
  35. A general regression model for lifetime evaluation and prediction (2001) — link
  36. Zipf distribution of us firm sizes (2001)
  37. Regular variation, subexponentiality and their applications in probability theory (1999)
  38. Signal processing with alpha-stable distributions and applications (1995)
  39. On the Self-Similar Nature of Ethernet Traffic ( extended version ) (1995)
  40. Strange kinetics (1993) — link
  41. The Ωμ+1-Rule (1981) — link
  42. Approximation Theorems of Mathematical Statistics (1980) — link
  43. The algebra of random variables (1979) — link
  44. A Theorem on Sums of Independent Positive Random Variables and Its Applications to Branching Random Processes (1964) — link
  45. Cours d’´economie politique , vol. 1 (1964)
  46. Mandelbrot and the Stable Paretian Hypothesis (1963) — link
  47. The Variation of Certain Speculative Prices (1963) — link
  48. Sturgeon’s law (1958)
  49. The Claustrophile (1956)
  50. The Frequency Distribution of the Quotient of Two Normal Variates — link
  51. THE RANDOM WALK:S GUIDE TO ANOMALOUS DIFFUSION: A FRACTIONAL DYNAMICS APPROACH

0 of 59 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.