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

Roberto Vila, Helton Saulo

Abstract

The Hoover index is a widely used measure of inequality with an intuitive interpretation, yet little is known about the finite-sample properties of its empirical estimator. In this paper, we derive a simple expression for the expected value of the Hoover index estimator for general non-negative populations, based on Laplace transform techniques and exponential tilting. This unified framework applies to both continuous and discrete distributions. Explicit bias expressions are obtained for gamma population, showing that the estimator is generally biased in finite samples. Numerical and simulation results illustrate the magnitude of the bias and its dependence on the underlying distribution and sample size.

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

Abstract

The Hoover index is a widely used measure of inequality with an intuitive interpretation, yet little is known about the finite-sample properties of its empirical estimator. In this paper, we derive a simple expression for the expected value of the Hoover index estimator for general non-negative populations, based on Laplace transform techniques and exponential tilting. This unified framework applies to both continuous and discrete distributions. Explicit bias expressions are obtained for gamma population, showing that the estimator is generally biased in finite samples. Numerical and simulation results illustrate the magnitude of the bias and its dependence on the underlying distribution and sample size.
Paper Structure (9 sections, 13 theorems, 70 equations, 3 figures)

This paper contains 9 sections, 13 theorems, 70 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Characterizations and special cases of the Hoover index
  3. Technical results
  4. Expectation of the Hoover index estimator
  5. Illustrative simulation study
  6. Concluding remarks
  7. Appendix
  8. Biased Hoover estimator in Poisson populations
  9. Biased Hoover estimator in geometric populations

Key Result

Proposition 2.1

The Hoover index of a non-negative, non-degenerate random variable $X$ with finite mean $\mathbb{E}[X]=\mu>0$ can be expressed as where $F(t^-)=\mathbb{P}(X<t)$.

Figures (3)

  • Figure 1: Lorenz curve illustrating the Hoover index and the Gini coefficient.
  • Figure 2: Relative bias of the Hoover estimator under the gamma distribution. Solid line: bias-corrected estimator; dashed line: original estimator.
  • Figure 3: RMSE of the Hoover estimator under the gamma distribution. Solid line: bias-corrected estimator; dashed line: original estimator.

Theorems & Definitions (33)

  • Definition 1.1
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • ...and 23 more