Estimation of inequality indices of the cumulative distribution function

Abstract

Inequality indices for self-assessed health and life satisfaction are typically constructed as functions of the cumulative distribution function. We present a unified methodology for the estimation of the resulting inequality indices. We also obtain explicit standard error formulas in the context of two popular families of inequality indices that have emerged from this literature.

Asymptotic Versus Bootstrap Inference for Inequality Indices of the Cumulative Distribution Function

“…As an illustrative example, we return to the data application in Abul Naga and Stapenhurst (2015) pertaining to asymptotic inference in relation to the Alphabeta family of inequality indices (3). These are data on five ordered nutritional health states from the Egyptian Integrated Household Survey of 1997-1999.…”

“…Inequality indices expressed as functions of the cumulative distribution function (CDF) are routinely used in studies that quantify inequality in self-assessed health, happiness, and other life satisfaction variables that are collected in the form of ordered response data. The large sample distribution of inequality indices of the CDF has been obtained by Abul Naga and Stapenhurst (2015). It is, however, an open question as to how reliable the adoption of the large sample distribution in testing hypotheses in applied work involving finite samples sizes is.…”

“…Here, V BOOT is the empirical variance of the B values ∆( X b ; j, ω 1 , ω 2 ) of the level of inequality. The sample quantiles of the B bootstrap statisticsz b are used instead of the quantiles of the standard 5 See Abul Naga and Stapenhurst (2015) for a derivation of a consistent estimator of the asymptotic variance of the inequality index. normal distribution in order to provide critical values for hypotheses tests related to the level of inequality in the underlying population.…”

Inferring inequality: Testing for median-preserving spreads in ordinal data

“…As Davidson and Duclos (2013), we adopt a likelihood ratio statistic combined with bootstrap inference, but we additionally consider both a Z statistic and asymptotic inference. Our work is also related to Abul Naga and Stapenhurst (2015) and Abul Naga et al (2020): while they perform inference on a random variable derived from a particular class of indices consistent with the MPS ordering, we perform inference on the binary outcome given by the partial ordering itself.…”