# On the estimation of the Lorenz curve under complex sampling designs

## TL;DR

This paper introduces a new estimator for the Lorenz curve under complex sampling designs, along with a resampling method for inference, enabling more accurate confidence intervals and tests for economic inequality measures.

## Contribution

It proposes a Hajek-type estimator for the Lorenz curve under complex sampling and develops a resampling scheme for inference, which was not previously available.

## Key findings

- The estimator is asymptotically normal.
- The resampling scheme accurately approximates the estimator's distribution.
- Simulation studies confirm the effectiveness of the proposed methods.

## Abstract

This paper focuses on the estimation of the concentration curve of a finite population, when data are collected according to a complex sampling design with different inclusion probabilities. A (design-based) Hajek type estimator for the Lorenz curve is proposed, and its asymptotic properties are studied. Then, a resampling scheme able to approximate the asymptotic law of the Lorenz curve estimator is constructed. Applications are given to the construction of (i) a confidence band for the Lorenz curve, (ii) confidence intervals for the Gini concentration ratio, and (iii) a test for Lorenz dominance. The merits of the proposed resampling procedure are evaluated through a simulation study.