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

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.

• The paper introduces the arithmetic-harmonic inequality index, a bounded, scale-invariant measure of multiplicative dispersion with explicit closed-form expressions.
• It develops a plug-in estimator with finite-sample analysis and validates theoretical results through extensive Monte Carlo simulations for GIG, IG, and Gamma distributions.
• The analysis establishes strong consistency, asymptotic normality, and bias quantification, showcasing the index's robustness in statistical modeling and economic inequality assessment.

The Arithmetic-Harmonic Inequality Index: Formal Summary

Definition and Theoretical Foundations

The arithmetic-harmonic inequality (AHI) index $J(X)$, introduced as $J = 1 - [\mathbb{E}[X]\mathbb{E}[1/X]]^{-1}$ for positive random variables $X$, provides a bounded, scale-invariant measure of multiplicative dispersion. The index exploits the fundamental relationship between the arithmetic mean $\mathbb{E}[X]$ and the harmonic mean $1/\mathbb{E}[1/X]$, and is strictly defined for distributions admitting finite first moments for both $X$ and $1/X$. From Jensen’s inequality, $J$ is bounded within $[0, 1)$, achieves $0$ only when $J = 1 - [\mathbb{E}[X]\mathbb{E}[1/X]]^{-1}$0 is degenerate, and is invariant under scaling and reciprocal transformations.

A second-order Taylor expansion establishes an approximate relationship between $J = 1 - [\mathbb{E}[X]\mathbb{E}[1/X]]^{-1}$1 and the squared coefficient of variation: $J = 1 - [\mathbb{E}[X]\mathbb{E}[1/X]]^{-1}$2 for small dispersion, with $J = 1 - [\mathbb{E}[X]\mathbb{E}[1/X]]^{-1}$3 as $J = 1 - [\mathbb{E}[X]\mathbb{E}[1/X]]^{-1}$4. Notably, for $J = 1 - [\mathbb{E}[X]\mathbb{E}[1/X]]^{-1}$5, the Atkinson index reduces to the AHI index, connecting it to prominent measures in inequality literature.

Distributional Results and Closed-form Expressions

Analytical results for $J = 1 - [\mathbb{E}[X]\mathbb{E}[1/X]]^{-1}$6 are derived within the generalized inverse Gaussian (GIG) family, whose density is parametrized by $J = 1 - [\mathbb{E}[X]\mathbb{E}[1/X]]^{-1}$7, $J = 1 - [\mathbb{E}[X]\mathbb{E}[1/X]]^{-1}$8, and $J = 1 - [\mathbb{E}[X]\mathbb{E}[1/X]]^{-1}$9; key moments are given in terms of modified Bessel functions $X$0. The population index in this family is:

$X$1

demonstrating dependence on $X$2 and $X$3 only.

Special cases are detailed:

• Inverse Gaussian: For $X$4, $X$5, $X$6, $X$7, establishing the dependence on the shape.
• Gamma: As $X$8 with $X$9, $\mathbb{E}[X]$0, $\mathbb{E}[X]$1, aligning exactly with $\mathbb{E}[X]$2 for this family.

Extension remarks articulate broader applicability to reciprocal and inverse Gamma, and Lévy distributions, but explicit computations are omitted.

Statistical Inference and Estimation

The plug-in estimator $\mathbb{E}[X]$3 is constructed from sample means:

$\mathbb{E}[X]$4

with several invariances preserved (scaling, permutations, reciprocals).

An exact integral representation for $\mathbb{E}[X]$5 leverages Laplace transforms, yielding:

$\mathbb{E}[X]$6

where $\mathbb{E}[X]$7 is the joint Laplace transform. For GIG, IG, and Gamma distributions, closed or tractable forms for $\mathbb{E}[X]$8 and the estimator bias are provided, with explicit dependence on the parameters and relevant Bessel and Gamma functions.

Large-Sample Properties

Strong consistency and asymptotic normality are established:

• Consistency: $\mathbb{E}[X]$9 for i.i.d. samples with finite means for $1/\mathbb{E}[1/X]$0 and $1/\mathbb{E}[1/X]$1.
• Normality: $1/\mathbb{E}[1/X]$2, with asymptotic variance

$1/\mathbb{E}[1/X]$3

highlighting dependence on both second-order moments and covariance structure.

First-order bias is quantified:

$1/\mathbb{E}[1/X]$4

with explicit forms for GIG, IG, and Gamma cases, including the limiting unbiasedness for the Lévy distribution.

Finite-Sample Performance

Monte Carlo simulations validate theoretical results for IG ($1/\mathbb{E}[1/X]$5) and Gamma ($1/\mathbb{E}[1/X]$6) distributions across varying sample sizes and parameter configurations. Empirical bias and mean squared error decrease with increasing $1/\mathbb{E}[1/X]$7, conforming to asymptotic predictions. The simulation design exhaustively explores parameter regimes to evaluate dispersion sensitivity.

Implications and Future Directions

The AHI index offers a bounded, interpretable alternative to unbounded dispersion metrics, preserving scale invariance. Its closed-form expressions across probability families facilitate parametric modeling of dispersion, including robust inequality measurement in economics and related fields. The developed estimators possess desirable large-sample properties and tractable bias expansions, indicating practical utility.

The theoretical developments invite further exploration of the estimator under mixture models, heavy-tailed distributions, and robust settings. As the index directly relates to classical measures (especially for Gamma), its adoption in inequality and risk quantification is natural. Future work may also analyze its role in parametric bootstrapping, bias correction, and generalized functional inference.

Conclusion

The paper systematically introduces, analyzes, and validates the arithmetic-harmonic inequality index as a measure of dispersion in positive-valued distributions. Analytical and inferential results are comprehensive, covering GIG-related families with strong numerical and asymptotic findings. The simulation study corroborates estimator behavior, and the methodology is extensible to broader applications in statistical modeling and economic inequality assessment.