Unifying the Hoover and Gini indices: Analytical, bias, and computational aspects

1. Problem Statement

Economic inequality is traditionally measured using two distinct classical indices:

• The Hoover Index ($H$): Measures inequality based on the mean absolute deviation from the mean. It represents the proportion of total income that must be redistributed to achieve equality.
• The Gini Coefficient ($G$): Measures inequality based on pairwise differences between individuals and is closely linked to the Lorenz curve.

While both quantify inequality, they emphasize different aspects of dispersion. The Hoover index focuses on deviations from the mean, whereas the Gini coefficient captures interpersonal differences. In distributions with high skewness or extreme values, these measures can yield divergent conclusions. The paper addresses the lack of a unified framework that allows researchers to continuously interpolate between these two perspectives to assess the sensitivity of inequality conclusions to the underlying definition of dispersion.

2. Methodology

The authors propose a new family of inequality indices, denoted as $I_\lambda$, defined for a random variable $X \geq 0$ with finite mean $\mu$ and independent copies $X_1, X_2$.

Definition of the Index:
For an interpolation parameter $\lambda \in [0, 1]$, the index is defined as:
$$I_\lambda = \frac{1}{2\mu} \mathbb{E} \left| (1 - \lambda)(X_1 - \mu) + \lambda(X_1 - X_2) \right|$$

• When $\lambda = 0$, $I_0 = H$ (Hoover index).
• When $\lambda = 1$, $I_1 = G$ (Gini coefficient).
• For intermediate $\lambda$, the index combines mean deviations and pairwise differences.

Theoretical Derivations:

• Integral Representation: The authors derive a general integral form for $I_\lambda$ involving distribution functions.
• Gamma Distribution Analysis: They specialize the index for Gamma distributions ($X \sim \text{Gamma}(\alpha, \beta)$), deriving closed-form expressions involving incomplete gamma functions.
• Estimator Properties: They analyze the "plug-in" estimator $\hat{I}_\lambda$, obtained by replacing the population mean $\mu$ with the sample mean $\bar{X}$ and expectations with sample averages.
• Bias Analysis: Using Laplace transform techniques and exponential tilting (Esscher transform), they derive a general expression for the expectation of the estimator. They further derive explicit formulas for the bias of the estimator under Gamma populations, utilizing the Generalized Hypoexponential (GHypo) distribution to handle sums of tilted variables.

3. Key Contributions

1. Unified Framework: Introduction of a continuous parametric family $I_\lambda$ that bridges the Hoover and Gini indices, satisfying key properties such as scale invariance, boundedness, and the Pigou–Dalton transfer principle.
2. Analytical Solutions: Derivation of explicit analytical representations for $I_\lambda$ under Gamma distributions, expressed via incomplete gamma functions.
3. Statistical Inference:
   • Establishment of a general expression for the expected value of the plug-in estimator $\hat{I}_\lambda$ for non-negative populations.
   • Derivation of exact bias formulas for the estimator under Gamma populations.
   • Proof that the estimator is unbiased for the Gini coefficient ($\lambda=1$) under Gamma distributions, consistent with existing literature.
4. Comparison with Linear Interpolation: The paper demonstrates that while a linear combination of the estimators ($\hat{J}_\lambda = (1-\lambda)\hat{H} + \lambda\hat{G}$) is mathematically simpler, the natural plug-in estimator $\hat{I}_\lambda$ exhibits smaller bias in finite samples.

4. Results

Theoretical Results:

• The index $I_\lambda$ is proven to be scale-invariant and bounded between 0 and the Gini coefficient.
• The bias of the estimator $\hat{I}_\lambda$ decreases as the sample size $n$ increases.
• The bias is generally larger for highly skewed distributions (small shape parameter $\alpha$).

Simulation Study (Monte Carlo):

• Setup: 1,000 replications under Gamma distributions with varying shape parameters ($\alpha \in \{0.5, 1, 2, 5, 10\}$), interpolation parameters ($\lambda \in \{0.25, 0.5, 0.75\}$), and sample sizes ($n \in \{10, \dots, 120\}$).
• Findings:
  • The estimator $\hat{I}_\lambda$ shows good finite-sample performance.
  • Both Mean Squared Error (MSE) and variance decrease systematically as $n$ increases.
  • The bias of $\hat{I}_\lambda$ is consistently lower than that of the competing linear estimator $\hat{J}_\lambda$ across all scenarios.
  • Performance degrades slightly with higher skewness (lower $\alpha$), which is expected for inequality measures.

Empirical Application:

• Applied to GDP per capita data (PPP, constant 2021 international dollars) for countries in the Americas.
• The study calculated $H \approx 0.229$ and $G \approx 0.329$.
• The proposed index provided a smooth, continuous transition between these values (e.g., $\hat{I}_{0.5} \approx 0.258$), demonstrating its utility as a flexible tool for sensitivity analysis.

5. Significance

This paper provides a rigorous statistical foundation for a unified approach to measuring economic inequality. Its significance lies in:

• Flexibility: Allowing researchers to move beyond the binary choice between Hoover and Gini indices, enabling sensitivity analysis regarding how much weight is placed on mean deviations versus pairwise differences.
• Statistical Robustness: Providing exact bias corrections and analytical tools for the Gamma distribution, a common model for income data.
• Practical Utility: Demonstrating that the proposed estimator outperforms simple linear combinations of existing estimators in terms of bias, making it a superior tool for empirical economic analysis, particularly in small-to-moderate sample sizes.

The work bridges the gap between theoretical inequality measurement and practical statistical inference, offering a more nuanced framework for policy analysis and economic development studies.