On the bias of the Gini coefficient estimator for zero-truncated Poisson distributions

TL;DR

The paper addresses bias in the Gini coefficient estimator when the population follows a zero-truncated Poisson distribution $X\sim\text{ZTP}(\lambda)$, a setting relevant for quantifying heterogeneity in infectious-disease transmission. It develops a closed-form bias expression by leveraging a sized-biased construction and related transform techniques, and introduces a bias-corrected estimator $\widehat{G}_{bc}$ that accounts for truncation and finite-sample effects. The main contributions are the explicit bias formula involving $R_1(F)$, $R_\infty(F)$, and the size-biased term, plus a practical correction method validated via Monte Carlo simulations showing substantial bias reduction with comparable MSE. These results enhance reliable inequality assessment in truncated count data and have direct relevance to analyzing superspreading phenomena in epidemiology, with potential extensions to other truncated or discrete inequality indices.

Abstract

This paper analyzes the Gini coefficient estimator for zero-truncated Poisson populations, revealing the presence of bias, and provides a mathematical expression for the bias, along with a bias-corrected estimator, which is evaluated using Monte Carlo simulation methods.

Figures (2)

• Figure 1: Relative bias of Gini estimators vs. sample size $n$, for different values of $\lambda$.
• Figure 2: Mean squared error of Gini estimators vs. sample size $n$, for different values of $\lambda$.

Theorems & Definitions (17)

• Definition 2.1
• Proposition 2.1
• proof
• Theorem 3.1
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Proposition 3.4
• proof