What is the formula to calculate the gini with sample weight

Question

I need your helps to explain how I can obtain the same result as this function does: gini(x, weights=rep(1,length=length(x))) http://cran.r-project.org/web/packages/reldist/reldist.pdf --> page 2. Gini

Let's say, we need to measure the inocme of the population N. To do that, we can divide the population N into K subgroups. And in each subgroup kth, we will take nk individual and ask for their income. As the result, we will get the "individual's income" and each individual will have particular "sample weight" to represent for their contribution to the population N. Here is example that I simply get from previous link and the dataset is from NLS

rm(list=ls())
cat("\014")

library(reldist)
data(nls);data
help(nls)

# Convert the wage growth from (log. dollar) to (dollar)
y <- exp(recent$chpermwage);y

# Compute the unweighted estimate
gini_y <-  gini(y)

# Compute the weighted estimate
gini_yw <- gini(y,w=recent$wgt)

> --- Here is the result----
> gini_y = 0.3418394

> gini_yw = 0.3483615

I know how to compute the Gini without WEIGHTS by my own code. Therefore, I would like to keep the command gini(y) in my code, without any doubts. The only thing I concerned is that the way gini(y,w) operate to obtain the result 0.3483615. I tried to do another calculation as follow to see whether I can come up with the same result as gini_yw. Here is another code that I based on CDF, Section 9.5, from this book: ‘‘Relative Distribution Methods in the Social Sciences’’ by Mark S. Handcock,

#-------------------------
# test how gini computes with the sample weights

z <- exp(recent$chpermwage) * recent$wgt
gini_z <- gini(z)

# Result gini_z = 0.3924161

As you see, my calculation gini_z is different from command gini(y, weights). If someone of you know how to build correct computation to obtain exactly gini_yw = 0.3483615, please give me your advices.

Thanks a lot friends.

Answer 1

function (x, weights = rep(1, length = length(x))) 
{
    ox <- order(x)
    x <- x[ox]
    weights <- weights[ox]/sum(weights)
    p <- cumsum(weights)
    nu <- cumsum(weights * x)
    n <- length(nu)
    nu <- nu/nu[n]
    sum(nu[-1] * p[-n]) - sum(nu[-n] * p[-1])
}

This is the source code for the function gini which can be seen by entering gini into the console. No parentheses or anything else.

EDIT: This can be done for any function or object really.