Gini's mean difference and variance as measures of finite populations scales

TL;DR

This paper explores Gini's mean difference as an alternative to variance for measuring scale in finite populations, especially in the presence of outliers, and compares their properties through theoretical analysis and simulations.

Contribution

It provides a comprehensive comparison of Gini's mean difference and variance, including asymptotic properties, estimation strategies, and the impact of outliers in finite population sampling.

Findings

Gini's mean difference has desirable robustness properties.

Asymptotic normality and Edgeworth expansions are established for both statistics.

Simulation results illustrate the theoretical properties and estimation strategies.

Abstract

We consider Gini's mean difference statistic as an alternative to the empirical variance in the settings of finite populations where simple random samples are drawn without replacement. In particular, we discuss specific (in the finite population context) estimation strategies for a scale of the population, related to the alternative statistic under possible presence of outliers in the data. The paper presents also a wide comparative survey of properties of the Gini mean difference statistic and the empirical variance. It includes asymptotic properties of both statistics: the asymptotic normality, one-term Edgeworth expansions and bootstrap approximations for Studentized versions of the statistics. An estimation of the variances and other parameters of the statistics is also in the study, where we exploit an auxiliary information on the population elements in the case of its availability. Theoretical results are illustrated with a simulation study.