Lorenz Curve
Computes the (empirical) ordinary and generalized Lorenz curve of a vector x
Lc(x, n = rep(1,length(x)), plot = FALSE)
x |
a vector containing non-negative elements. |
n |
a vector of frequencies, must be same length as |
plot |
logical. If TRUE the empirical Lorenz curve will be plotted. |
Lc(x) computes the empirical ordinary Lorenz curve of x
as well as the generalized Lorenz curve (= ordinary Lorenz curve *
mean(x)). The result can be interpreted like this: p*100 percent
have L(p)*100 percent of x.
If n is changed to anything but the default x is
interpreted as a vector of class means and n as a vector of
class frequencies: in this case Lc will compute the minimal
Lorenz curve (= no inequality within each group). A maximal curve can be
computed with Lc.mehran.
A list of class "Lc" with the following components:
p |
vector of percentages |
L |
vector with values of the ordinary Lorenz curve |
L.general |
vector with values of the generalized Lorenz curve |
B C Arnold: Majorization and the Lorenz Order: A Brief Introduction, 1987, Springer,
F A Cowell: Measurement of Inequality, 2000, in A B Atkinson / F Bourguignon (Eds): Handbook of Income Distribution, Amsterdam,
F A Cowell: Measuring Inequality, 1995 Prentice Hall/Harvester Wheatshef.
## Load and attach income (and metadata) set from Ilocos, Philippines data(Ilocos) attach(Ilocos) ## extract and rescale income for the provinces "Pangasinan" und "La Union" income.p <- income[province=="Pangasinan"]/10000 income.u <- income[province=="La Union"]/10000 ## compute the Lorenz curves Lc.p <- Lc(income.p) Lc.u <- Lc(income.u) ## it can be seen the the inequality in La Union is higher than in ## Pangasinan because the respective Lorenz curve takes smaller values. plot(Lc.p) lines(Lc.u, col=2) ## the picture becomes even clearer with generalized Lorenz curves plot(Lc.p, general=TRUE) lines(Lc.u, general=TRUE, col=2) ## inequality measures emphasize these results, e.g. Atkinson's measure ineq(income.p, type="Atkinson") ineq(income.u, type="Atkinson") ## or Theil's entropy measure ineq(income.p, type="Theil", parameter=0) ineq(income.u, type="Theil", parameter=0) # income distribution of the USA in 1968 (in 10 classes) # x vector of class means, n vector of class frequencies x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261) n <- c(482, 825, 722, 690, 661, 760, 745, 2140, 1911, 1024) # compute minimal Lorenz curve (= no inequality in each group) Lc.min <- Lc(x, n=n) # compute maximal Lorenz curve (limits of Mehran) Lc.max <- Lc.mehran(x,n) # plot both Lorenz curves in one plot plot(Lc.min) lines(Lc.max, col=4) # add the theoretic Lorenz curve of a Lognormal-distribution with variance 0.78 lines(Lc.lognorm, parameter=0.78) # add the theoretic Lorenz curve of a Dagum-distribution lines(Lc.dagum, parameter=c(3.4,2.6))
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