Inference for the Lorenz Curve and Gini Index under the Geometric Distribution

Technical Summary: Inference for the Lorenz Curve and Gini Index under the Geometric Distribution

Problem Statement
Measures of inequality, specifically the Lorenz curve and the Gini index, are foundational tools in economics, reliability theory, and actuarial science. While their theoretical properties and inferential frameworks are well-established for continuous distributions (e.g., exponential, lognormal, Pareto), their application to discrete data presents significant analytical challenges. In discrete settings, the cumulative distribution function is a step function, rendering the standard integral definition of the Lorenz curve invalid. Consequently, existing literature lacks a systematic inferential theory for these measures under discrete parametric models. This paper addresses this gap by developing the exact and asymptotic distributional properties of maximum likelihood estimators (MLEs) for the Lorenz curve and Gini index under the geometric distribution, a fundamental model for count data in fields such as reliability, queuing, and epidemiology.

Methodology
The authors formulate a discrete analogue of the Lorenz curve by replacing integration with summation over the support of the probability mass function. For a geometric random variable $X \sim \text{Geom}(p)$ with support $\{0, 1, 2, \dots\}$, the Lorenz curve $L(u; p)$ is defined via the generalized inverse of the distribution function.

The study proceeds through the following methodological steps:

1. Estimator Derivation: Utilizing the invariance property of MLE, the estimators for the Lorenz curve ($\hat{L}(u)$) and the Gini index ($\hat{G}$) are derived directly from the MLE of the parameter $p$, denoted as $\hat{p} = n / (n + \sum X_i)$.
2. Exact Distribution Theory: Recognizing that the sufficient statistic $T = \sum X_i$ follows a negative binomial distribution, the authors derive the exact sampling distributions of $\hat{L}(u)$ and $\hat{G}$ in closed form. These distributions are lattice-based and non-smooth, reflecting the discrete nature of the underlying model.
3. Asymptotic Analysis: The paper establishes strong consistency and $L_k$-convergence for both estimators. Asymptotic normality is derived using the delta method, yielding explicit variance expressions.
4. Simulation and Application: A Monte Carlo simulation study (5,000 replications) evaluates finite-sample performance (bias, variance, MSE) across various parameter values ($p$) and sample sizes ($n$). The methodology is further illustrated using the Federalist Papers word count dataset.

Key Contributions and Results

• Exact Distributional Forms: The paper provides the first closed-form expressions for the exact sampling distributions of Lorenz curve and Gini index estimators under the geometric distribution.

  • For the Gini index, the estimator $\hat{G} = (n+T)/(2n+T)$ is shown to be a one-to-one function of the sufficient statistic $T$. Its distribution function is expressed via the incomplete negative binomial cumulative distribution function (CDF).
  • For the Lorenz curve, the distribution is derived based on the transformation of the negative binomial variable, revealing a stepwise structure dependent on the generalized inverse.

• Asymptotic Properties:

  • Gini Index: The estimator $\hat{G}$ is shown to be strongly consistent and asymptotically normal. The asymptotic variance is derived explicitly as $\tau^2 = p^2(1-p)/(2-p)^4$. Notably, the Gini index estimator does not require regularity conditions regarding the parameter $u$ (as it is a scalar functional of $p$), allowing for reliable inference even at moderate sample sizes.
  • Lorenz Curve: The estimator $\hat{L}(u)$ is consistent, but its asymptotic normality requires a regularity condition: the mapping $p \mapsto F^{-1}(u; p)$ must be constant in a neighborhood of the true parameter. Without this, the derivative is undefined.

• Finite-Sample Performance (Simulation Findings):

  • The Gini index estimator performs robustly; simulated variances closely track the asymptotic approximation even for small sample sizes ($n=20$).
  • In contrast, the Lorenz curve estimator exhibits significantly slower convergence to the asymptotic regime. Due to the discrete "jump" structure of the distribution, the estimator frequently crosses jump points of the generalized inverse in finite samples, inflating variance beyond the delta method's prediction. This effect is particularly pronounced for larger values of $p$. The simulation indicates that asymptotic normality for $\hat{L}(u)$ may require substantially larger sample sizes to be practically reliable.

• Real Data Application:

  • The methodology was applied to the Federalist Papers word count dataset (frequency of the word "may"). The geometric distribution provided an excellent fit (Pearson $\chi^2$ p-value = 0.783).
  • The estimated Gini index was $\hat{G} = 0.7162$ with a 95% confidence interval of $(0.6926, 0.7398)$, indicating high concentration in word usage.
  • The fitted Lorenz curve correctly captured the discrete feature where the lowest 40% of text blocks (those with zero occurrences) contribute nothing to the total count, resulting in a flat segment at the origin.

Significance and Claims
The paper claims to provide a rigorous inferential framework for inequality measures in discrete settings, extending classical Lorenz-based analysis to a fundamental discrete count model. The authors emphasize that while the Gini index estimator allows for standard asymptotic inference from moderate sample sizes, the Lorenz curve estimator necessitates the use of the derived exact distributions for accurate inference, particularly in smaller samples or when $p$ is large. This distinction highlights a fundamental difference between continuous and discrete inequality estimation. The work addresses a specific gap in the literature where systematic inferential theory (covering exact distributions, consistency, and asymptotic normality) for discrete parametric models was previously absent.