scholarly journals Group averaging and the Gini deviation

2021 ◽  
Vol 29 (3) ◽  
pp. 595-605
Author(s):  
Oleg I. Pavlov ◽  
Olga Yu. Pavlova
Keyword(s):  
Gini Coefficient ◽  
Upper Bound ◽  
Lorenz Curve ◽  
Piecewise Linear ◽  
Geometric Parameters ◽  
Natural Question ◽  
Lorenz Curves ◽  
The Gini Coefficient ◽  
The Difference ◽  
Group Averaging

It is known that partitioning a society into groups with subsequent averaging in each group decreases the Gini coefficient. The resulting Lorenz function is piecewise linear. This study deals with a natural question: by how much the Gini coefficient could decrease when passing to a piecewise linear Lorenz function? Obtained results are quite illustrative (since they are expressed in terms of the geometric parameters of the polygon Lorenz curve, such as the lengths of its segments and the angles between successive segments) upper bound estimates for the maximum possible change in the Gini coefficient with a restriction on the group shares, or on the difference between the averaged values of the attribute for consecutive groups. It is shown that there exist Lorenz curves with the Gini coefficient arbitrarily close to one, and at the same time with the Gini coefficient of the averaged society arbitrarily close to zero.

scholarly journals A note on a property of the Gini coefficient

2019 ◽  
Vol 27 (2) ◽  
pp. 81-88
Author(s):  
Marian Genèev
Keyword(s):  
Mathematical Method ◽  
Czech Republic ◽  
Gini Coefficient ◽  
Upper Bound ◽  
The Czech Republic ◽  
Gini Coefficients ◽  
The Gini Coefficient ◽  
The Difference

AbstractThe scope of this note is a self-contained presentation of a mathematical method that enables us to give an absolute upper bound for the difference of the Gini coefficients|G(σ1, . . ., σn) − G(γ1, . . . , γn)| ,where (γ1, . . . , γn) represents the vector of the gross wages and (σ1, . . . , σn) represents the vector of the corresponding super-gross wages that is used in the Czech Republic for calculating the net wage. Since (as of June 2019) σi = 100 ⎡ 1.34 γi/100⎤, the study of the above difference seems to be somewhat inaccessible for many economists. However, our estimate based on the presented technique implies that the introduction of the super-gross wage concept does not essentially affect the value of the Gini coefficient as sometimes expected.

Anything Lorenz Curves Can Do, Top Shares Can Do: Assessing the TopBot Family of Inequality Measures

2018 ◽  
Vol 49 (4) ◽  
pp. 947-981 ◽  
Author(s):  
Guillermina Jasso
Keyword(s):  
Gini Coefficient ◽  
Probability Distributions ◽  
The United States ◽  
Current Evidence ◽  
Inequality Measure ◽  
Lorenz Curves ◽  
Inequality Measures ◽  
Top Incomes ◽  
The Gini Coefficient ◽  
And Shifts

Newly precise evidence of the trajectory of top incomes in the United States and around the world relies on shares and ratios, prompting new inquiry into their properties as inequality measures. Current evidence suggests a mathematical link between top shares and the Gini coefficient and empirical links extending as well to the Atkinson measure. The work reported in this article strengthens that evidence, making several contributions: First, it formalizes the shares and ratios, showing that as monotonic transformations of each other, they are different manifestations of a single inequality measure, here called TopBot. Second, it presents two standard forms of TopBot, which satisfy the principle of normalization. Third, it presents a new link between top shares and the Gini coefficient, showing that properties and results associated with the Lorenz curve pertain as well to top shares. Fourth, it investigates TopBot in mathematically specified probability distributions, showing that TopBot is monotonically related to classical measures such as the Gini, Atkinson, and Theil measures and the coefficient of variation. Thus, TopBot appears to be a genuine inequality measure. Moreover, TopBot is further distinguished by its ease of calculation and ease of interpretation, making it an appealing People’s measure of inequality. This work also provides new insights, for example, that, given nonlinearities in the (monotonic) relations among inequality measures, Spearman correlations are more appropriate than Pearson correlations and that weakening of correlations signals differences and shifts in distributional form, themselves signals of income dynamics.

Lorenz Curve Interpolation and the Gini Coefficient

2010 ◽  
Author(s):  
Nicholas Charles Rohde
Keyword(s):  
Gini Coefficient ◽  
Lorenz Curve ◽  
Polynomial Spline ◽  
The Philippines ◽  
The Body ◽  
Grouped Data ◽  
Parametric Methods ◽  
Lorenz Curves ◽  
Curve Interpolation ◽  
Income Decile

This article presents a simple non-polynomial spline that may be used to construct Lorenz curves from grouped data. The spline is naturally convex and works by determining a series of piecewise segments that may be joined to give a smooth and continuous Lorenz curve. The method is illustrated with an empirical example using income decile data from the Philippines from 1991-2003 where the proposed technique is used alongside other parametric and non-parametric methods. We also use the spline to approximate some known Lorenz curves and assess the technique by comparing the estimated Gini coefficient to the known Gini. Our findings suggest that the method is an attractive addition to the body of techniques used for developing Lorenz curves from grouped data.

scholarly journals Intersecting Lorenz curves and aversion to inverse downside inequality

2020 ◽  
Author(s):  
W. Henry Chiu
Keyword(s):  
Gini Coefficient ◽  
Linear Inequality ◽  
The Other ◽  
Inequality Aversion ◽  
Inequality Index ◽  
Lorenz Curves ◽  
Income Distributions ◽  
The Gini Coefficient ◽  
Precise Condition

Abstract This paper defines and characterizes the concept of an increase in inverse downside inequality and show that, when the Lorenz curves of two income distributions intersect, how the change from one distribution to the other is judged by an inequality index exhibiting inverse downside inequality aversion often depends on the relative strengths of its aversion to inverse downside inequality and inequality aversion. For the class of linear inequality indices, of which the Gini coefficient is a member, a measure characterizing the strength of an index’s aversion to inverse downside inequality against its own inequality aversion is shown to determine the ranking by the index of two distributions whose Lorenz curves cross once. The precise condition under which the same result generalizes to the case of multiple-crossing Lorenz curves is also identified.

A Study of the Estimation of the Gini Coefficient of Income Using Lorenz Curve

2016 ◽  
Vol 15 (4) ◽  
pp. 1-10 ◽  
Author(s):  
Kwasi Darkwah ◽  
Ezekiel Nortey ◽  
Felix Mettle ◽  
Isaac Baidoo
Keyword(s):  
Gini Coefficient ◽  
Lorenz Curve ◽  
The Gini Coefficient

Some Variations on Standard Income Measurement Error Models

2004 ◽  
Author(s):  
Brigham B. Frandsen ◽  
James B. McDonald
Keyword(s):  
Measurement Error ◽  
Gini Coefficient ◽  
Measurement Error Models ◽  
The Gini Coefficient ◽  
Parametric Specification ◽  
Income Measurement ◽  
The Difference ◽  
Multiplicative Measurement Error ◽  
The Relationship ◽  
Measures Of Inequality

Measurement error can have a significant impact on measures of inequality. Using a fairly flexible parametric specification of an independent multiplicative measurement error (IMME) model we explore the relationship between changes in the variance of measurement error, for a given mean of measurement error, on the Gini Coefficient. While the measured Gini is greater than the true Gini, the difference decreases as the variance of measurement error decreases. Copulas are used to relax the assumption of independence of measurement error and true income. In this case the measured Gini can be larger or smaller than the true Gini, depending on the correlation between true income and measurement error. Using the same approach with simulations the effect of a different distribution of measurement error is investigated.

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