giniVarCI (version 0.0.1-3)

fginindex: Gini index for finite populations and different estimation methods.

Description

Estimates the Gini index in finite populations, using different methods.

Usage

fginindex(
 y,
 w,
 method = 2L,
 Pi = NULL,
 na.rm = TRUE,
 useRcpp = TRUE
)

Value

A single numeric value between 0 and 1. The estimation of the Gini index.

Arguments

y

A vector with the non-negative real numbers to be used for estimating the Gini index.

w

A numeric vector with the survey weights to be used for estimating the Gini index. This argument can be missing if argument Pi is provided.

method

An integer between 1 and 5 selecting one of the 5 methods detailed below for estimating the Gini index in finite populations. The default method is method = 2L.

Pi

A numeric vector with the (sample) first inclusion probabilites to be used for estimating the Gini index. This argument can be NULL if argument w is provided. The default value is Pi = NULL.

na.rm

A `TRUE/FALSE` logical value indicating whether NA's should be removed before the computation proceeds. The default value is na.rm = TRUE.

useRcpp

A `TRUE/FALSE` logical value indicating whether Rcpp (useRcpp = TRUE), or R (useRcpp = FALSE), is used for computation. The default value is UseRcpp = TRUE.

Author

Juan F Munoz

Jose M Pavia

Encarnacion Alvarez

Details

For a sample $S$, with size $n$ and inclusion probabilities ${\pi }_i=P(i\in S)$ (argument Pi), derived from a finite population $U$, with size $N$, different formulations of the Gini index have been proposed in the literature. This function estimates the Gini index using various formulations, and both R and C++ codes are implemented. This can be useful for research purposes, and speed comparisons can be made. The different methods for estimating the Gini index are (see also Muñoz et al., 2023):

method = 1 (Langel and Tillé, 2013)

$${\hat{G}}_{w1}={\displaystyle \frac{1}{2{\hat{N}}^2{\bar{y}}_w}\sum _{i\in S}\sum _{j\in S}{w}_i{w}_j|y_i-y_j|,}$$ where $\hat{N}=\sum _{i\in S}{w}_i$, ${\bar{y}}_w={\hat{N}}^{-1}\sum _{i\in S}{w}_i{y}_i$, and $w_i$ are the survey weights. For example, the survey weights can be $w_i={\pi }_i^{-1}$. w or Pi must be provided, but not both. It is required that $w_i={\pi }_i^{-1}$, for $i\in S$, when both w and Pi are provided.

method = 2 (Alfons and Templ, 2012; Langel and Tillé, 2013)

$${\hat{G}}_{w2}={\displaystyle \frac{2\sum _{i\in S}{w}_{(i)}^{\ast }{\hat{N}}_{(i)}{y}_{(i)}-\sum _{i\in S}{w}_i^2{y}_i}{{\hat{N}}^2{\bar{y}}_w}-1,}$$ where $y_{(i)}$ are the values $y_i$ sorted in increasing order, $w_{(i)}^{\ast }$ are the values $w_i$ sorted according to the increasing order of the values $y_i$, and ${\hat{N}}_{(i)}=\sum _{j=1}^i{w}_{(j)}^{\ast }$. Langel and Tillé (2013) show that ${\hat{G}}_{w1}={\hat{G}}_{w2}$.

method = 3 (Berger, 2008)

$${\hat{G}}_{w3}={\displaystyle \frac{2}{\hat{N}{\bar{y}}_w}\sum _{i\in S}{w}_i{y}_i{\hat{F}}_w^{\ast }(y_i)-1,}$$ where $${\hat{F}}_w^{\ast }(t)={\displaystyle \frac{1}{\hat{N}}\sum _{i\in S}{w}_i[\delta (y_i<t)+0.5\delta (y_i=t)]}$$ is the smooth (mid-point) distribution function, and $\delta (\cdot )$ is the indicator variable that takes the value 1 when its argument is true, and the value 0 otherwise. It can be seen that ${\hat{G}}_{w2}={\hat{G}}_{w3}$.

method = 4 (Berger and Gedik-Balay, 2020)

$${\hat{G}}_{w4}=1-{\displaystyle \frac{{\bar{z}}_w}{{\bar{y}}_w},}$$ where ${\bar{z}}_w={\hat{N}}^{-1}\sum _{i\in S}{w}_i{z}_i$ and $$z_i={\displaystyle \frac{1}{\hat{N}-w_i}\sum _{\begin{array}{c}j\in S\\ j\ne i\end{array}}min(y_i,y_j).}$$

method = 5 (Lerman and Yitzhaki, 1989)

$${\hat{G}}_{w5}={\displaystyle \frac{2}{\hat{N}{\bar{y}}_w}\sum _{i\in S}{w}_i[y_i-{\bar{y}}_w][{\hat{F}}_w^{LY}(y_i)-{\bar{F}}_w^{LY}],}$$ where $${\hat{F}}_w^{LY}(y_i)={\displaystyle \frac{1}{\hat{N}}({\hat{N}}_{(i-1)}+\frac{w_{(i)}^{\ast }}{2})}$$ and ${\bar{F}}_w^{LY}={\hat{N}}^{-1}\sum _{i\in S}{w}_i{\hat{F}}_w^{LY}(y_i)$.

References

Alfons, A., and Templ, M. (2012). Estimation of social exclusion indicators from complex surveys: The R package laeken. KU Leuven, Faculty of Business and Economics Working Paper.

Berger, Y. G. (2008). A note on the asymptotic equivalence of jackknife and linearization variance estimation for the Gini Coefficient. Journal of Official Statistics, 24(4), 541-555.

Berger, Y. G., and Gedik-Balay, İ. (2020). Confidence intervals of Gini coefficient under unequal probability sampling. Journal of official statistics, 36(2), 237-249.

Langel, M., and Tillé, Y. (2013). Variance estimation of the Gini index: revisiting a result several times published. Journal of the Royal Statistical Society: Series A (Statistics in Society), 176(2), 521-540.

Lerman, R. I., and Yitzhaki, S. (1989). Improving the accuracy of estimates of Gini coefficients. Journal of econometrics, 42(1), 43-47.

Muñoz, J. F., Moya-Fernández, P. J., and Álvarez-Verdejo, E. (2023). Exploring and Correcting the Bias in the Estimation of the Gini Measure of Inequality. Sociological Methods & Research. https://doi.org/10.1177/00491241231176847

See Also

fgini, fcompareCI

Examples

# Income and weights (region "Burgenland") from the 2006 Austrian EU-SILC (Package 'laeken').
data(eusilc, package="laeken")
y <- eusilc$eqIncome[eusilc$db040 == "Burgenland"]
w <- eusilc$rb050[eusilc$db040 == "Burgenland"]

#Comparing the computation time for the various estimation methods and using R
microbenchmark::microbenchmark(
fginindex(y, w, method = 1L,  useRcpp = FALSE),
fginindex(y, w, method = 2L,  useRcpp = FALSE),
fginindex(y, w, method = 3L,  useRcpp = FALSE),
fginindex(y, w, method = 4L,  useRcpp = FALSE),
fginindex(y, w, method = 5L,  useRcpp = FALSE)
)

# Comparing the computation time for the various estimation methods and using Rcpp
microbenchmark::microbenchmark(
fginindex(y, w, method = 1L),
fginindex(y, w, method = 2L),
fginindex(y, w, method = 3L),
fginindex(y, w, method = 4L),
fginindex(y, w, method = 5L)
)



# Estimation of the Gini index using 'method = 4'.
y <- c(30428.83, 14976.54, 18094.09, 29476.79, 20381.93, 6876.17,
       10360.96, 8239.82, 29476.79, 32230.71)
w <- c(357.86, 480.99, 480.99, 476.01, 498.58, 498.58, 476, 498.58, 476.01, 476.01)
fginindex(y, w, method = 4L)