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Insights into Tail-Based and Order Statistics

Hamidreza Maleki Almani

TL;DR

The paper addresses quantile-contribution statistics for heavy-tailed distributions by linking them to order statistics. It develops closed-form, exact distributional results for tail-based statistics in small samples and establishes almost-sure convergence of the tail statistic to a function of the $p$-quantile and the mean. It then derives the asymptotic distribution of the numerator and, via a Geary–Hinkley transformation, the asymptotic distribution of the tail-based statistic itself, offering a lognormal approximation for practical use. Comprehensive simulations across diverse distributions corroborate the theoretical results and demonstrate the approach's utility for analyzing heavy-tailed data. These contributions provide a rigorous, implementable framework for applying quantile contributions to risk, inequality, and related tail phenomena.

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.

Insights into Tail-Based and Order Statistics

TL;DR

The paper addresses quantile-contribution statistics for heavy-tailed distributions by linking them to order statistics. It develops closed-form, exact distributional results for tail-based statistics in small samples and establishes almost-sure convergence of the tail statistic to a function of the -quantile and the mean. It then derives the asymptotic distribution of the numerator and, via a Geary–Hinkley transformation, the asymptotic distribution of the tail-based statistic itself, offering a lognormal approximation for practical use. Comprehensive simulations across diverse distributions corroborate the theoretical results and demonstrate the approach's utility for analyzing heavy-tailed data. These contributions provide a rigorous, implementable framework for applying quantile contributions to risk, inequality, and related tail phenomena.

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.

Paper Structure

This paper contains 7 sections, 15 theorems, 105 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Ordered and Tail-Based Statistics
  3. Exact Distribution of Tail-Based Statistics
  4. Convergence of Tail-Based Statistics
  5. Asymptotic Distribution of Numerator
  6. Asymptotic Distribution of Tail-Based Statistic
  7. Simulation and Callibration

Key Result

Proposition 2.1

The probability distribution $F_{(i)}^n$ and density function $f_{(i)}^n$ of the order statistic $X_{(i)}^n$ are where $F$ and $f$ are respectively the probability distribution and density function of the variable $X_1$, and $B,I$ are respectively the beta function and the regularized inclomplete beta function, i.e., for all $\mathfrak{Re}(p),\mathfrak{Re}(q)>0$

Figures (2)

  • Figure 1: The histogram, analytical log-normal probability density function, and estimated probability density function of $\Lambda_n$ for i.i.d variables from different continuous distributions.
  • Figure :

Theorems & Definitions (27)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof : Proof 1: Calculus
  • proof : Proof 2: Combinatorics
  • Corollary 2.6
  • ...and 17 more