Insights into Tail-Based and Order Statistics

TL;DR

This paper provides a theoretical analysis of quantile contribution statistics in heavy-tailed distributions, deriving explicit formulas, studying asymptotic behavior, and validating results through simulations for small and large samples.

Contribution

It introduces a closed-form joint distribution for order statistics and quantile contributions, and characterizes their asymptotic properties in heavy-tailed contexts.

Findings

Explicit CDF for quantile contributions in small samples

Asymptotic normality of numerator in quantile contributions

Simulation results confirm theoretical convergence and accuracy

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.