Lorenz asymmetry coefficient

The Lorenz asymmetry coefficient ( asymmetry coefficient ) is a parameter of the Lorenz curve that measures the degree of asymmetry of the curve .

definition

This is defined as:

${\displaystyle S=F(\mu )+L(\mu ),}$

where the functions and are defined as in the Lorenz curve and is the arithmetic mean. If is, then the point at which the Lorenz curve runs parallel to the perfect line of perfect equality is above the axis of symmetry. Accordingly, the point at which the Lorenz curve is parallel to the perfect straight line of equality is below the axis of symmetry. ${\displaystyle F}$ ${\displaystyle L}$ ${\displaystyle \mu }$ ${\displaystyle S>1}$ ${\displaystyle S<1}$

If the data come from a logarithmic normal distribution , then is , that is, the Lorenz curve is symmetrical. ${\displaystyle S=1}$

The sample parameter can be calculated from the ordered data sets using the following equations: ${\displaystyle S}$ ${\displaystyle n}$ ${\displaystyle \left(x_{1},\ldots ,x_{m},x_{m+1},\ldots ,x_{n}\right)}$

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

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

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

where the number of individuals with a size is less than and . ${\displaystyle m}$ ${\displaystyle \mu }$ ${\displaystyle L_{i}=\sum _{j=1}^{i}x_{j}}$

literature

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

Individual evidence

1. ^ ^{A b} Christian Damgaard, Jacob Weiner: Describing inequality in plant size or fecundity. 4th ed. 81. Vol. Ecology, 2000. doi : 10.1890 / 0012-9658 (2000) 081 [1139: DIIPSO] 2.0.CO; 2 . Pp. 1139-1142.