Lorenz asymmetry coefficient

The Lorenz asymmetry coefficient is a summary statistic of the Lorenz curve that measures the degree of asymmetry of the curve. The Lorenz asymmetry coefficient is defined as

$S=F(\mu )+L(\mu )\,$

where the functions F and L are defined as for the Lorenz curve, and μ is the mean. If S > 1, then the point where the Lorenz curve is parallel with the line of equality is above the axis of symmetry. Correspondingly, if S < 1, then the point where the Lorenz curve is parallel to the line of equality is below the axis of symmetry.

If data arise from the log-normal distribution then S = 1, i.e., the Lorenz curve is symmetric.

The sample statistic S can be calculated from n ordered size data, $(x_{1},...,x_{m},x_{m+1},...,x_{n})$, using the following equations:

$\delta ={\frac {\mu -x_{m}}{x_{m+1}-x_{m}}}$

$F(\mu )={\frac {m+\delta }{n}}$

$L(\mu )={\frac {L_{m}+\delta x_{m+1}}{L_{n}}}$,

where m is the number of individuals with a size less than μ.

References

• Damgaard, Christian; Weiner, Jacob (2000). "Describing inequality in plant size or fecundity". Ecology. 81 (4): 1139–1142. doi:10.1890/0012-9658(2000)081[1139:DIIPSO]2.0.CO;2.

External links

• LORENZ 3.0 is a Mathematica notebook which draw sample Lorenz curves and calculates Gini coefficients and Lorenz asymmetry coefficients from data in an Excel sheet.