Parametric estimation of quantile versions of Zenga and D inequality curves: methodology and application to Weibull distribution

Inequality (concentration) curves such as Lorenz, Bonferroni, Zenga curves, as well as a new inequality curve – the D curve, are broadly used to analyse inequalities in wealth and income distribution in certain populations. Quantile versions of these inequality curves are more robust to outliers. We discuss several parametric estimators of quantile versions of the Zenga and D curves. A minimum distance (MD) estimator is proposed for these two curves and the indices related to them. The consistency and asymptotic normality of the MD estimator is proved. The MD estimator can also be used to estimate the inequality measures corresponding to the quantile versions of the inequality curves. The estimation methods considered are illustrated in the case of the Weibull model, which has many applications in life sciences, for example, to fit the precipitation data. In econometrics it is also considered to fit incomes, especially in the case when a significant share of population have low incomes, for example, in less developed countries or among low-paid jobs.