Linearization of inequality indices in the design-based framework

Abstract

Linearization methods are customarily adopted in sampling surveys to obtain approximated variance formulae for estimators of statistical functionals under the design-based approach. In the present paper, following the Deville [Variance estimation for complex statistics and estimators: linearization and residual techniques. Surv Methodol. 1999;25:193â\u80\u93203] approach stemming from the concept of design-based influence function, we provide a general result for linearizing a large family of population functionals which includes many of the inequality measures considered in social, economic and statistical studies, such as the Gini, Amato, Zenga, Atkinson and Generalized Entropy indices. The feasibility of our theoretical results is assessed by some simulation studies involving real and artificial data

On the estimation of the Lorenz curve under complex sampling designs

“…The main consequences of Proposition 7 are two. First of all, the estimator S 2 * is a consistent estimator of the variance of R H ; variance estimation based on linearization techniques is dealt with, for instance, in [Barabesi et al(2016)]. In the second place, the confidence intervals…”

“…The estimation of inequality measures (mainly Gini's index), when data are collected according to a variable probability sampling design from a finite population, is widely studied in the literature: cfr. [Langel and Tillé(2013)], [Barabesi et al(2016)], and references therein.…”

A critical issue of using the variance of the total in the linearization method – In the context of unequal probability sampling

“…Langel and Tillé obtained the variance of Gini Index (an indicator of income inequality) using various approaches including those proposed by Deville, Demnati and Rao, and Graf, where a total of linearized variables are primarily calculated. This type of the linearization technique is continuously used for variance estimation and found in many most recent publications (eg, the works of Barabesi et al, Dudel, and Bellettiere et al).…”