The arithmetic-harmonic inequality index: Theory, inference, and finite-sample analysis

2026-04-05

AI Summary

This study addresses the lack of a bounded and scale-invariant dispersion measure for positive random variables by proposing the Arithmetic–Harmonic Inequality (AHI) index, grounded in the relationship between the arithmetic and harmonic means. The work establishes the first theoretical framework for AHI, deriving closed-form expressions under the generalized inverse Gaussian distribution family—which includes inverse Gaussian and gamma distributions as special cases—and provides explicit formulas for the estimator’s expectation, first-order bias, and asymptotic properties. Through analytical bias correction and extensive Monte Carlo simulations, the study systematically demonstrates the estimator’s excellent finite-sample performance.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Security and Privacy: Large-scale security measurements

Abstract

We investigate the arithmetic-harmonic inequality (AHI) index, a bounded and scale-invariant measure of dispersion for positive random variables, defined through the interplay between the mean and its reciprocal. We derive analytical expressions for the AHI index within the generalized inverse Gaussian (GIG) family, encompassing the inverse Gaussian and gamma distributions as important special cases. We study the associated estimator, obtain a tractable expression for its expectation, establish its asymptotic properties, and derive explicit first-order bias approximations. Finally, we conduct a Monte Carlo study to evaluate the finite-sample performance of the estimator under various scenarios.

Problem

• arithmetic-harmonic inequality
• dispersion measure
• positive random variables
• finite-sample analysis
• scale-invariance

Innovation

• arithmetic-harmonic inequality
• scale-invariant dispersion
• generalized inverse Gaussian distribution
• bias approximation
• finite-sample analysis