🤖 AI Summary
This paper investigates the intrinsic relationship between quantile contribution statistics and order statistics under heavy-tailed distributions. Addressing the challenge of precisely characterizing extreme-value contributions in small samples, we derive, for the first time, a closed-form expression for the joint distribution of order statistics, enabling an explicit cumulative distribution function for quantile contributions. We further establish asymptotic normality of this statistic under large-sample conditions and characterize its limiting distributional properties. Our methodology integrates order statistics theory, extreme-value analysis, asymptotic inference, and Monte Carlo simulation. Key contributions are: (1) an exact finite-sample distributional characterization of extreme-value contributions; (2) a systematic analysis of asymptotic behavior and convergence rates; and (3) a theoretically grounded, computationally tractable, and interpretable framework applicable to financial risk modeling, network anomaly detection, and biological extreme-value analysis.
📝 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.