On the bias of the Hoover index estimator: Results for the gamma distribution

1. Problem Statement

The Hoover index (also known as the Robin Hood, Pietra, or Schutz index) is a widely used measure of inequality defined as half the relative mean absolute deviation from the mean. While the Gini coefficient has been extensively studied regarding its finite-sample properties and estimator bias, the statistical behavior of the empirical Hoover index estimator remains under-researched.

Existing literature largely relies on asymptotic results or numerical investigations for the Hoover index. There is a lack of exact, closed-form expressions for the finite-sample bias of the Hoover index estimator, particularly for specific parametric families like the Gamma distribution, which is commonly used to model income and wealth data. This paper aims to fill that gap by deriving exact expressions for the expected value and bias of the estimator.

2. Methodology

The authors employ a unified analytical framework based on Laplace transform techniques and exponential tilting (also known as Esscher transformation).

• General Framework: The paper first establishes a general expression for the expected value of the Hoover index estimator ($\hat{H}$) for any non-negative, non-degenerate random variable. This is achieved by:
  1. Expressing the estimator using an integral representation involving the Laplace transform of the sample sum.
  2. Utilizing the identity $z \int_0^\infty e^{-zx} dx = 1$ to transform the expectation of a ratio into an integral of expectations.
  3. Applying Lemma 3.1, which provides a formula for the expectation of the absolute difference of two independent variables weighted by exponential terms, expressed in terms of exponentially tilted distributions.
• Specialization to Gamma Distribution: The general result is then specialized for populations following a Gamma distribution ($X \sim \text{Gamma}(\alpha, \lambda)$). The authors leverage the properties of the Gamma distribution under exponential tilting (which results in another Gamma distribution with a modified rate parameter) to derive a closed-form integral expression for the expected value.
• Bias Correction: Based on the derived analytical bias, a bias-corrected estimator ($\hat{H}_c$) is constructed using a plug-in approach, where the shape parameter $\alpha$ is estimated via Maximum Likelihood (ML).
• Simulation: A Monte Carlo study is conducted to validate the theoretical findings, comparing the uncorrected estimator against the bias-corrected version across various sample sizes ($n$) and shape parameters ($\alpha$).

3. Key Contributions

The paper makes several significant theoretical and practical contributions:

• Unified Expected Value Expression: The authors derive a general closed-form representation for $E[\hat{H}]$ (Theorem 4.1) applicable to both continuous and discrete distributions. This expression is formulated in terms of the Laplace transform of the population and the cumulative distribution functions (CDFs) of exponentially tilted random variables.
• Explicit Bias for Gamma Populations: A specific, tractable expression for the bias of the Hoover index estimator under the Gamma distribution is derived (Corollary 4.5). The bias is shown to depend on the sample size $n$ and the shape parameter $\alpha$, but is scale-invariant (independent of the rate parameter $\lambda$).
• Connection to Gini Coefficient: The paper establishes a theoretical link for the case $n=2$, showing that the expected value of the Hoover index estimator is exactly half the expected value of the Gini coefficient estimator (Proposition 4.8).
• Bias Correction Mechanism: The derivation allows for the construction of a bias-corrected estimator that removes the systematic finite-sample error without significantly increasing variance.

4. Key Results

• Existence of Bias: The study confirms that the empirical Hoover index estimator is biased in finite samples. Specifically, for Gamma distributions, the estimator exhibits a negative bias (it underestimates the true inequality).
• Magnitude of Bias: The bias is most pronounced for small sample sizes and specific values of the shape parameter $\alpha$. As the sample size $n$ increases, the bias diminishes, consistent with asymptotic consistency.
• Performance of Correction:
  • Relative Bias: Simulation results (Figure 2) demonstrate that the bias-corrected estimator effectively eliminates the negative bias, with relative bias values fluctuating around zero even for moderate sample sizes ($n=25$).
  • Efficiency: The Root Mean Squared Error (RMSE) of the bias-corrected estimator is nearly identical to that of the original uncorrected estimator (Figure 3). This indicates that the bias correction improves accuracy without sacrificing efficiency or increasing variability.
• Bounds: The paper provides theoretical bounds for the expected value and the bias (Remarks 4.3 and 4.7).

5. Significance and Implications

• Empirical Reliability: The findings are crucial for researchers and policymakers using the Hoover index in fields like economics, demography, and environmental studies. The results suggest that using the raw estimator in small-to-moderate samples can lead to a systematic underestimation of inequality.
• Practical Application: The proposed bias-correction method is computationally feasible (plug-in ML estimation) and highly effective, offering a simple way to improve the accuracy of inequality measurements.
• Theoretical Foundation: By linking the Hoover index to Laplace transforms and exponential tilting, the paper opens new avenues for analyzing inequality measures. The authors suggest future research directions involving connections to skew-normal and skew-elliptical distributions, similar to recent developments in Gini coefficient analysis, which could further enhance the understanding of the Hoover index's sampling properties.

In conclusion, this paper provides the first exact finite-sample analysis of the Hoover index estimator, proving its inherent bias in Gamma populations and offering a robust, efficient solution for bias correction.