Editing Pareto distribution (section)

===Lorenz curve and Gini coefficient===
[[File:ParetoLorenzSVG.svg|thumb|325px|Lorenz curves for a number of Pareto distributions. The case ''α''&nbsp;=&nbsp;∞ corresponds to perfectly equal distribution (''G''&nbsp;=&nbsp;0) and the ''α''&nbsp;=&nbsp;1 line corresponds to complete inequality (''G''&nbsp;=&nbsp;1)]]

The [[Lorenz curve]] is often used to characterize income and wealth distributions. For any distribution, the Lorenz curve ''L''(''F'') is written in terms of the PDF ''f'' or the CDF ''F'' as

:<math>L(F)=\frac{\int_{x_\mathrm{m}}^{x(F)}xf(x)\,dx}{\int_{x_\mathrm{m}}^\infty xf(x)\,dx} =\frac{\int_0^F x(F')\,dF'}{\int_0^1 x(F')\,dF'}</math>

where ''x''(''F'') is the inverse of the CDF. For the Pareto distribution,

:<math>x(F)=\frac{x_\mathrm{m}}{(1-F)^{\frac{1}{\alpha}}}</math>

and the Lorenz curve is calculated to be

:<math>L(F) = 1-(1-F)^{1-\frac{1}{\alpha}},</math>

For <math>0<\alpha\le 1</math> the denominator is infinite, yielding ''L''=0. Examples of the Lorenz curve for a number of Pareto distributions are shown in the graph on the right.

According to [[Oxfam]] (2016) the richest 62 people have as much wealth as the poorest half of the world's population.<ref>{{cite web|title=62 people own the same as half the world, reveals Oxfam Davos report|url=https://www.oxfam.org/en/pressroom/pressreleases/2016-01-18/62-people-own-same-half-world-reveals-oxfam-davos-report|publisher=Oxfam|date=Jan 2016}}</ref> We can estimate the Pareto index that would apply to this situation. Letting ε equal <math>62/(7\times 10^9)</math> we have:
:<math>L(1/2)=1-L(1-\varepsilon)</math>
or
:<math>1-(1/2)^{1-\frac{1}{\alpha}}=\varepsilon^{1-\frac{1}{\alpha}}</math>
<!--:<math>\ln(1-(1/2)^{1-\frac{1}{\alpha}})=(1-\frac{1}{\alpha})\ln\varepsilon</math>
:<math>\ln(1-(1/2)^{1-\frac{1}{\alpha}})=(\ln\varepsilon/\ln 2)(1-\frac{1}{\alpha})\ln 2</math>
:<math>\ln(1-(1/2)^{1-\frac{1}{\alpha}})=-(\ln\varepsilon/\ln 2)\ln((1/2)^{1-\frac{1}{\alpha}})</math>
:<math>\ln(1-(1/2)^{1-\frac{1}{\alpha}})\approx(\ln\varepsilon/\ln 2)(1-(1/2)^{1-\frac{1}{\alpha}})</math>
:<math>-\ln(1-(1/2)^{1-\frac{1}{\alpha}})\exp(-\ln(1-(1/2)^{1-\frac{1}{\alpha}}))\approx -\ln\varepsilon/\ln 2</math>
:<math>-\ln(1-(1/2)^{1-\frac{1}{\alpha}})\approx W(-\ln\varepsilon/\ln 2)</math>
where ''W'' is the [[Lambert W function]]. So
:<math>(1/2)^{1-\frac{1}{\alpha}}\approx 1-\exp(-W(-\ln\varepsilon/\ln 2))</math>
:<math>{1-\frac{1}{\alpha}}\approx -\ln(1-\exp(-W(-\ln\varepsilon/\ln 2)))/\ln 2</math>
:<math>\alpha\approx 1/(1+\ln(1-\exp(-W(-\ln\varepsilon/\ln 2)))/\ln 2)</math>
-->The solution is that ''α'' equals about 1.15, and about 9% of the wealth is owned by each of the two groups. But actually the poorest 69% of the world adult population owns only about 3% of the wealth.<ref>{{cite web|title=Global Wealth Report 2013|url=https://publications.credit-suisse.com/tasks/render/file/?fileID=BCDB1364-A105-0560-1332EC9100FF5C83|publisher=Credit Suisse|page=22|date=Oct 2013|access-date=2016-01-24|archive-url=https://web.archive.org/web/20150214155424/https://publications.credit-suisse.com/tasks/render/file/?fileID=BCDB1364-A105-0560-1332EC9100FF5C83|archive-date=2015-02-14|url-status=dead}}</ref>

The [[Gini coefficient]] is a measure of the deviation of the Lorenz curve from the equidistribution line which is a line connecting [0,&nbsp;0] and [1,&nbsp;1], which is shown in black (''α''&nbsp;=&nbsp;∞) in the Lorenz plot on the right. Specifically, the Gini coefficient is twice the area between the Lorenz curve and the equidistribution line. The Gini coefficient for the Pareto distribution is then calculated (for <math>\alpha\ge 1</math>) to be

:<math>G = 1-2 \left (\int_0^1L(F) \, dF \right ) = \frac{1}{2\alpha-1}</math>

(see Aaberge 2005).