# Inferential results for a new measure of inequality

## TL;DR

This paper introduces a new inequality index tailored to detect significant changes in both tails of income distributions, providing estimators with proven asymptotic properties and applying them to real Italian income data.

## Contribution

It proposes a novel inequality measure, develops two estimators, and establishes their asymptotic properties, including consistency and normality, with application to real income data.

## Key findings

- Estimators are asymptotically equivalent.
- The proposed estimator is consistent and asymptotically normal.
- Application to Italian income data demonstrates practical relevance.

## Abstract

In this paper we derive inferential results for a new index of inequality, specifically defined for capturing significant changes observed both in the left and in the right tail of the income distributions. The latter shifts are an apparent fact for many countries like US, Germany, UK, and France in the last decades, and are a concern for many policy makers. We propose two empirical estimators for the index, and show that they are asymptotically equivalent. Afterwards, we adopt one estimator and prove its consistency and asymptotic normality. Finally we introduce an empirical estimator for its variance and provide conditions to show its convergence to the finite theoretical value. An analysis of real data on net income from the Bank of Italy Survey of Income and Wealth is also presented, on the base of the obtained inferential results.