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Langel, Matti
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Langel, Matti
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- PublicationMétadonnées seulementVariance estimation of the Gini index: Revisiting a result several times published(2013)
; Since Corrado Gini suggested the index that bears his name as a way of measuring inequality, the computation of variance of the Gini index has been subject to numerous publications. In this paper, we survey a large part of the literature related to the topic and show that the same results, as well as the same errors, have been republished several times, often with a clear lack of reference to previous work. Whereas existing literature on the subject is very fragmented, we regroup papers from various fields and attempt to bring a wider view of the problem. Moreover, we try to explain how this situation occurred and the main issues involved when trying to perform inference on the Gini index, especially under complex sampling designs. The interest of several linearization methods is discussed and the contribution of recent papers is evaluated. Also, a general result to linearize a quadratic form is given, allowing the approximation of variance to be computed in only a few lines of calculation. Finally, the relevance of the regression-based approach is evaluated and an empirical comparison is proposed. - PublicationMétadonnées seulementHistogram-based interpolation of the Lorenz curve and Gini index for grouped data(2012-10-1)
; In grouped data, the estimation of the Lorenz curve without taking into account the within-class variability leads to an overestimation of the curve and an underestimation of the Gini index. We propose a new strictly convex estimator of the Lorenz curve derived from a linear interpolation-based approximation of the cumulative distribution function. Integrating the Lorenz curve, a correction can be derived for the Gini index which takes the intra-class variability into account. - PublicationAccès libreInference by linearization for Zenga’s new inequality index: a comparison with the Gini index(2012-9-17)
; Zenga’s new inequality curve and index are two recent tools for measuring inequality. Proposed in 2007, they should thus not be mistaken for anterior measures suggested by the same author. This paper focuses on the new measures only, which are hereafter referred to simply as the Zenga curve and Zenga index. The Zenga curve Z (alpha) involves the ratio of the mean income of the 100 alpha% poorest to that of the 100(1-alpha)% richest. The Zenga index can also be expressed by means of the Lorenz Curve and some of its properties make it an interesting alternative to the Gini index. Like most other inequality measures, inference on the Zenga index is not straightforward. Some research on its properties and on estimation has already been conducted but inference in the sampling framework is still needed. In this paper, we propose an estimator and variance estimator for the Zenga index when estimated from a complex sampling design. The proposed variance estimator is based on linearization techniques and more specifically on the direct approach presented by Demnati and Rao. The quality of the resulting estimators are evaluated in Monte Carlo simulation studies on real sets of income data. Finally, the advantages of the Zenga index relative to the Gini index are discussed. - PublicationAccès libreMeasuring inequality in finite population sampling(2012)
; Ce document se concentre sur l’estimation des mesures d’inégalité à l’aide de données d’enquête. La méthodologie proposée permet de tenir compte du caractère non-linéaire des mesures d’inégalité ainsi que de la complexité de la stratégie d’échantillonnage. Le premier chapitre est dédié à la présentation et à la définition des concepts principaux de l’étude quantitative des inégalités et de la théorie des sondages. Dans le second chapitre, plusieurs indices d’inégalité sont comparés au sein d’une étude empirique réalisée à l’aide de données réelles. La recherche se centre ensuite vers trois mesures d’inégalités spécifiques : le Quintile share ratio (QSR), l’indice de Gini et l’indice de Zenga. Ainsi, dans le troisième chapitre, nous montrons que la variance du QSR peut être estimée par linéarisation sans avoir recours à un lissage par noyau et qu’une simple transformation permet d’améliorer le taux de couverture de l’intervalle de confiance. Les deux chapitres suivants abordent les travaux de Corrado Gini sous un angle particulier, notamment à travers des réflexions historiques sur l’échantillonnage équilibré dont il a été l’un des pionniers, et sur l’estimation de variance de l’indice d’inégalité qui porte son nom. L’ultime chapitre est dédié à la présentation d’une mesure moins connue, l’indice de Zenga, pour laquelle nous proposons un estimateur de variance., This document focuses on the estimation of inequality measures for complex survey data. The proposed methodology takes into account both the complexity of these generally non-linear functions of interest and the complexity of the sampling strategy. The first chapter is dedicated to the presentation and definition of the main concepts used in both inequality and survey sampling theory. In the second chapter, a variety of inequality indices are compared in an empirical study on a real set of income data. Research is then directed towards three specific inequality measures: the Quintile share ratio (QSR), the Gini index and Zenga’s new inequality index. The third chapter shows that the variance of the QSR can be estimated by means of the linearization approach without applying a kernel smoothing, and that a simple transformation enhances the coverage rate of the confidence interval. The two following chapters discuss the work of Corrado Gini from an unusual angle. For instance, both balanced sampling (of which he is a pioneer) and variance estimation for the inequality measure that bears his name are discussed in a historical perspective. Zenga’s new inequality index is presented in the last chapter and a variance estimator is proposed. - PublicationAccès libreInference by linearization for Zenga’s new inequality index: a comparison with the Gini index(2012)
; Zenga’s new inequality curve and index are two recent tools for measuring inequality. Proposed in 2007, they should thus not be mistaken for anterior measures suggested by the same author. This paper focuses on the new measures only, which are hereafter referred to simply as the Zenga curve and Zenga index. The Zenga curve Z(α) involves the ratio of the mean income of the 100 α %poorest to that of the 100 (1-α)% richest. The Zenga index can also be expressed by means of the Lorenz Curve and some of its properties make it an interesting alternative to the Gini index. Like most other inequality measures, inference on the Zenga index is not straightforward. Some research on its properties and on estimation has already been conducted but inference in the sampling framework is still needed. In this paper, we propose an estimator and variance estimator for the Zenga index when estimated from a complex sampling design. The proposed variance estimator is based on linearization techniques and more specifically on the direct approach presented by Demnati and Rao. The quality of the resulting estimators are evaluated in Monte Carlo simulation studies on real sets of income data. Finally, the advantages of the Zenga index relative to the Gini index are discussed. - PublicationMétadonnées seulement
- PublicationMétadonnées seulement
- PublicationAccès libreCorrado Gini, a pioneer in balanced sampling and inequality theory(2011-3-14)
; This paper attempts to make the link between two of Corrado Gini’s contributions to statistics: the famous inequality measure that bears his name and his work in the early days of balanced sampling. Some important notions of the history of sampling such as representativeness, randomness, and purposive selection are clarified before balanced sampling is introduced. The Gini index is described, as well as its estimation and variance estimation in the sampling framework. Finally, theoretical grounds and some simulations on real data show how some well used auxiliary information and balanced sampling can enhance the accuracy of the estimation of the Gini index. - PublicationMétadonnées seulement
- PublicationMétadonnées seulement