lorenzdominancerelation {TUGLab} | R Documentation |
Lorenz dominance relation
Description
Given two awards vectors, this function returns the Lorenz dominance relation between them.
Usage
lorenzdominancerelation(x, y)
Arguments
x |
A vector. |
y |
A vector. |
Details
In order to compare two vectors x,y\in \mathbb{R}^n
through the Lorenz criterion,
both of them must be rearranged in non-decreasing order; thus, let \bar{x}
and \bar{y}
be the vectors obtained by rearranging x
and y
, respectively, in non-decreasing order.
It is said that x
Lorenz-dominates y
(or that y
is Lorenz-dominated by x
)
if all the cumulative sums of \bar{x}
are not less than those of \bar{y}
.
That is, x
Lorenz-dominates y
if \sum_{j=1}^{n}\bar{x}_j=\sum_{j=1}^{n}\bar{y}_j
and, for each k=1,\dots,n-1
,
\sum_{j=1}^{k}\bar{x}_j \geq \sum_{j=1}^{k}\bar{y}_j.
If x
Lorenz-dominates y
and y
Lorenz-dominates x
,
then x
and y
are said to be Lorenz-equal.
If x
does not Lorenz-dominate y
and y
does not Lorenz-dominate x
,
then x
and y
are not Lorenz-comparable.
Value
There are four possible outputs:
-1 |
if the introduced vectors are not Lorenz-comparable. |
0 |
if the vectors are Lorenz-equal. |
1 |
if the vectors are not Lorenz-equal and the first one Lorenz-dominates the second one. |
2 |
if the vectors are not Lorenz-equal and the second one Lorenz-dominates the first one. |
References
Lorenz, M. O. (1905). Methods of Measuring the Concentration of Wealth. Publications of the American Statistical Association, 9(70), 209-219.
Examples
lorenzdominancerelation(c(1,2,3), c(1,1,4))
lorenzdominancerelation(c(1,2,7,2), c(1,1,4,6))