Interval Estimation for Robust Versions of Lorenz Curve IntegratingEmpirical Likelihood Framework with Influence Functions

Abstract

Real-world data often pose significant challenges for statistical inference due to issues like outliers, violation of classical assumptions, and inherent heterogeneity. Traditional parametric methods, heavily reliant on assumptions such as normality and homoscedasticity, frequently yield unreliable results when applied to skewed distributions, exemplified by income data with extremely high earners. Empirical likelihood methods have gained prominence to overcome these limitations as they reduce dependence on strict distributional assumptions. Further robustness is achieved by incorporating influence functions into empirical likelihood frameworks, explicitly quantifying estimator sensitivity to data contamination, and improving inference reliability. The Lorenz curve is a crucial economic measure of income and wealth disparity, but is sensitive to skewness and outliers. This study introduces robust quantile-based estimators for the Lorenz curve by integrating empirical likelihood and influence function methodologies. Confidence intervals are constructed through empirical likelihood optimization using a Lagrange multiplier framework and are evaluated against three bootstrap methods. Extensive simulations reflecting realistic income distributions demonstrate that the influence function-based empirical likelihood method significantly outperforms traditional empirical likelihood and other statistical approaches. Its superior robustness arises primarily from effectively mitigating the distortive effects of extreme observations. Consequently, this approach ensures reliable inference even under challenging real-world conditions characterized by data abnormalities. Nevertheless, the proposed methods present limitations, notably computational complexity and sensitivity to the assumptions underlying influence functions. Future research should expand its application across socioeconomic contexts, improve computational efficiency, and deepen theoretical and empirical exploration. This framework equips researchers and policymakers with precise tools to measure and analyze income inequality.