# Description of methods for estimating Gini indexes, variance estimates and confidence intervals using GiniVarInterval

The aim of this vignette is to describe the various methods for estimating the Gini index, for both infinite and finite populations, as well as the methods for estimating its variance, as implemented in the giniVarCI package. Different confidence intervals for the Gini index are also explained.

To exemplify the use of the different functions, we assume that inequality is measured for a nonnegative and continuous random variable $$Y$$. A popular formulation of the Gini index ($$G$$) is defined by (see David, 1968; Kendall and Stuart, 1977; Qin et al., 2010): $G = \frac{1}{2 \mu_{Y}} \int_{0}^{+\infty} \int_{0}^{+\infty} |x-y|dF_{Y}(x)dF_{Y}(y),$ where $\mu_{Y}=E[Y]=\int_{0}^{+\infty}yf(y)dy=\int_{0}^{+\infty}ydF_{Y}(y),$ is the mean of $$Y$$, and $$F_{Y}(y)=P(Y\leq y)$$ and $$f(y)$$ are the cumulative distribution function and the probability density function of $$Y$$, respectively.

In practice, the value of $$G$$ is estimated by means of a sample $$S$$ with size $$n$$, which can be selected from either infinite or finite populations (Berger and Gedik Balay, 2020; Muñoz et al., 2023).

# 1 Infinite populations

## 1.1 Estimators of the Gini index

For infinite populations, $$\{Y_{i}: i\in S\}$$ denotes a sequence, with size $$n$$, of nonnegative random variables with the same distribution as the variable of interest $$Y$$. The Gini index ($$G$$) is estimated using the observation of individuals selected in the sample, which are denoted as $$\{y_{i}: i\in S\}$$. A popular estimator of the Gini index is (see Langel and Tille, 2013; Giorgi and Gigliarano, 2017; Muñoz et al., 2023): $\widehat{G} = \displaystyle \frac{2}{\overline{y}n^2}\sum_{i \in S}iy_{(i)} - \frac{n+1}{n},$ where $$\overline{y}=n^{-1}\sum_{i=1}^{n}y_i$$, and $$y_{(i)}$$ are the ordered values (in non-decreasing order) of the sample observations $$y_i$$. This is the expression computed by the functions iginindex() (method = 5) and igini() when bias.correction = FALSE.

The estimator $$\widehat{G}$$ can be biased for small sample sizes (Deltas, 2003). The bias corrected (bc) version of $$\widehat{G}$$ is: $\widehat{G}^{bc} = \displaystyle \frac{2}{\overline{y}n(n-1)}\sum_{i \in S}iy_{(i)} - \frac{n+1}{n-1},$ which corresponds to the Gini index bias correction version computed by iginindex() (method = 5) and igini() when bias.correction = TRUE.

In the first example, a sample with size n=100 is generated using the gsample() function from the standard logNormal distribution (distribution = "lognormal") with true Gini index is $$G=0.5$$ (gini = 0.5) and the Gini index is estimated using bias correction.

library(giniVarCI)
set.seed(123)
y <- gsample(n = 100, gini = 0.5, distribution = "lognormal")
igini(y)
#> [1] 0.4671929

iginindex() can be used to estimate the Gini index using various estimation methods and both R and C++ codes. See help(iginindex) for a detailed description of the various estimation methods. Efficiency comparisons between both implementations and with other functions available in other packages, such as laeken, DescTools, ineq or REAT, can be made using, for instance, the function microbenchmark():



#Comparing the computation time for the various estimation methods using R
microbenchmark::microbenchmark(
iginindex(y, method = 1,  useRcpp = FALSE),
iginindex(y, method = 2,  useRcpp = FALSE),
iginindex(y, method = 3,  useRcpp = FALSE),
iginindex(y, method = 4,  useRcpp = FALSE),
iginindex(y, method = 5,  useRcpp = FALSE),
iginindex(y, method = 6,  useRcpp = FALSE),
iginindex(y, method = 7,  useRcpp = FALSE),
iginindex(y, method = 8,  useRcpp = FALSE),
iginindex(y, method = 9,  useRcpp = FALSE),
iginindex(y, method = 10, useRcpp = FALSE)
)
#> Unit: microseconds
#>                                        expr    min       lq      mean   median
#>   iginindex(y, method = 1, useRcpp = FALSE)  144.1   164.10   256.703   185.15
#>   iginindex(y, method = 2, useRcpp = FALSE)   13.9    20.30    33.951    25.50
#>   iginindex(y, method = 3, useRcpp = FALSE)   11.5    17.10    29.406    21.55
#>   iginindex(y, method = 4, useRcpp = FALSE)   15.5    22.85    36.819    28.70
#>   iginindex(y, method = 5, useRcpp = FALSE)   16.2    21.80    41.324    27.45
#>   iginindex(y, method = 6, useRcpp = FALSE)   31.3    63.60    95.618    80.15
#>   iginindex(y, method = 7, useRcpp = FALSE)  948.3  1024.75  1356.977  1149.25
#>   iginindex(y, method = 8, useRcpp = FALSE)  779.4   897.05  1148.337   975.20
#>   iginindex(y, method = 9, useRcpp = FALSE)  751.7   832.75  1073.753   916.95
#>  iginindex(y, method = 10, useRcpp = FALSE) 9899.3 12344.85 14473.612 14318.45
#>        uq     max neval
#>    239.50  3925.8   100
#>     39.05   154.9   100
#>     31.75   215.0   100
#>     43.95   170.7   100
#>     41.70   544.4   100
#>    102.00   602.7   100
#>   1499.85  4320.1   100
#>   1254.35  4180.3   100
#>   1275.00  3225.7   100
#>  15689.05 24814.1   100

# Comparing the computation time for the various estimation methods using Rcpp
microbenchmark::microbenchmark(
iginindex(y, method = 1),
iginindex(y, method = 2),
iginindex(y, method = 3),
iginindex(y, method = 4),
iginindex(y, method = 5),
iginindex(y, method = 6),
iginindex(y, method = 7),
iginindex(y, method = 8),
iginindex(y, method = 9),
iginindex(y, method = 10) )
#> Unit: microseconds
#>                       expr     min       lq      mean   median       uq     max
#>   iginindex(y, method = 1)    21.2    27.00    58.562    36.20    49.85  1068.6
#>   iginindex(y, method = 2)    10.0    15.50    53.593    20.90    26.50  2465.1
#>   iginindex(y, method = 3)     9.9    17.35    33.202    23.95    32.75   287.1
#>   iginindex(y, method = 4)    10.2    18.15    41.034    24.20    30.70  1275.7
#>   iginindex(y, method = 5)     9.0    12.95    22.263    20.50    25.35   146.6
#>   iginindex(y, method = 6)     8.4    13.85    23.954    19.65    24.90   126.1
#>   iginindex(y, method = 7)    31.0    46.85    70.911    58.30    67.65   442.9
#>   iginindex(y, method = 8)    20.5    27.75    47.885    36.80    44.65   480.6
#>   iginindex(y, method = 9)    19.0    29.75    66.249    40.65    55.00   940.7
#>  iginindex(y, method = 10) 10681.2 16687.45 25121.255 22552.65 30578.30 87119.2
#>  neval
#>    100
#>    100
#>    100
#>    100
#>    100
#>    100
#>    100
#>    100
#>    100
#>    100


# Comparing the computation time for estimates of the Gini index in various R packages.

microbenchmark::microbenchmark(
igini(y),
laeken::gini(y),
DescTools::Gini(y),
ineq::Gini(y),
REAT::gini(y))
#> Registered S3 methods overwritten by 'DescTools':
#>   method   from
#>   lines.Lc ineq
#>   plot.Lc  ineq
#> Unit: microseconds
#>                expr   min     lq      mean median     uq       max neval
#>            igini(y)   9.8  12.00    34.814  14.35  18.05    1864.2   100
#>     laeken::gini(y)  35.3  38.00   392.260  41.20  50.95   32288.5   100
#>  DescTools::Gini(y)  56.6  61.80 11304.144  67.20  78.40 1122424.1   100
#>       ineq::Gini(y)  45.7  49.45   133.872  52.60  65.90    6572.8   100
#>       REAT::gini(y) 100.7 108.30   234.544 113.10 137.15    8623.5   100

## 1.2 Variance estimation and confidence intervals

Variance estimators and confidence intervals are described using different methods for the estimator of the non-bias corrected version of Gini index $$\widehat{G}$$, since as $\widehat{G}^{bc} = \frac{n}{n-1}\widehat{G},$ the variance estimators and confidence intervals based on $$\widehat{G}^{bc}$$ can be straightforwardly derived. In particular, $\widehat{V}(\widehat{G}^{bc})=\frac{n^2}{(n-1)^2}\widehat{V}(\widehat{G}).$ Let $$[L,U]$$ the lower and upper limits of a confidence interval for $$G$$ based on $$\widehat{G}$$. The confidence interval based on $$\widehat{G}^{bc}$$ can be computed as: $\left[ \frac{n}{n-1}L, \frac{n}{n-1}U\right].$

### 1.2.1 Bootstrap

The argument interval = pbootstrap in the function igini() returns the confidence interval for the Gini index using the percentile bootstrap method. Let $$\{y_{1}^{*}(b),\ldots, y_{n}^{*}(b)\}$$ be the $$b$$th bootstrap sample taken from the original sample $$\{y_{1},\ldots, y_{n}\}$$ by simple random sampling with replacement, and $$\widehat{G}^{*}(b)$$ denotes the estimator $$\widehat{G}$$ computed from the $$b$$th bootstrap sample, with $$b=\{1,\ldots,B\}$$, being $$B$$ the total number of bootstrap samples. For a confidence level $$1-\alpha$$, the percentile bootstrap confidence interval is defined as (see Qin et al., 2010): $\left[ \widehat{G}^{*}_{(\alpha/2)}, \widehat{G}^{*}_{(1-\alpha/2)} \right],$ where $$\widehat{G}^{*}_{(a)}$$ is the $$a$$th quantile of the bootstrapped coefficients $$\widehat{G}^{*}(b)$$. A variance estimator of the Gini index based on bootstrap is defined as $\widehat{V}_{B}(\widehat{G})= \displaystyle \frac{1}{B-1}\sum_{b=1}^{B}\left(\widehat{G}^{*}(b) - \overline{G}^{*} \right)^2,$ where $\overline{G}^{*}=\frac{1}{B}\sum_{b=1}^{B}\widehat{G}^{*}(b).$


# Gini index estimation and confidence interval using 'pbootstrap',

igini(y, interval = "pbootstrap")
#> $Gini #> [1] 0.4671929 #> #>$Interval
#>          lower     upper
#> [1,] 0.4004204 0.5135315
#>
#> $Variance #> [1] 0.0008333577 interval = "BCa" computes the bias-corrected and accelerated bootstrap interval (Davison and Hinkley, 1997). The idea of this confidence interval is to correct for bias due to the skewness in the distribution of bootstrap estimates. The "BCa" confidence interval is defined as: $\left[ \widehat{G}^{*}_{(\alpha_{1})}, \widehat{G}^{*}_{(\alpha_{2})} \right],$ where $\alpha_{1}=\phi\left( \widehat{Z}_{0} + \frac{\widehat{Z}_{0} + Z_{\alpha}}{1-\widehat{a} (\widehat{Z}_{0} + \widehat{Z}_{\alpha}) } \right),$ $\alpha_{2}=\phi\left( \widehat{Z}_{0} + \frac{\widehat{Z}_{0} + Z_{1-\alpha}}{1-\widehat{a} (\widehat{Z}_{0} + \widehat{Z}_{1-\alpha}) } \right),$ $$\phi(\cdot)$$ is the cumulative distribution function of the standard Normal distribution, and $$Z_{a}$$ is the $$a$$th quantile of the standard Normal distribution. The bias-correction factor is defined as $\widehat{Z}_{0}=\phi^{-1}\left(\#\frac{\widehat{G}^{*}(b) - \widehat{G}}{B}\right),$ and the acceleration factor is given by $\widehat{a}=\frac{\sum_{i \in S}\left(\overline{G} -\widehat{G}_{-i} \right)^3}{6\left\{\sum_{i \in S}\left(\overline{G} -\widehat{G}_{-i} \right)^2\right\}^{3/2}},$ where $$\widehat{G}_{-i}$$ are the jackknife estimates defined in the following section, and $\overline{G} = \frac{1}{n}\sum_{i \in S}\widehat{G}_{-i}.$  # Gini index estimation and confidence interval using 'Bca'. igini(y, interval = "BCa") #>$Gini
#> [1] 0.4671929
#>
#> $Interval #> lower upper #> [1,] 0.4178437 0.5280127 #> #>$Variance
#> [1] 0.0008051247

### 1.2.2 Jackknife

The "zjackknife" and "tjackknife" methods compute the variance of the Gini index using the Ogwang Jackknife procedure (Ogwang, 2000; Langel and Tille, 2013). This variance si given by $\widehat{V}_{J}(\widehat{G})= \displaystyle \frac{n-1}{n}\sum_{i \in S}\left(\widehat{G}_{-i}- \overline{G} \right)^2,$ where $\widehat{G}_{-i}=\widehat{G}+\frac{2}{\sum_{j \in S}y_j - y_{(i)} }\left[ \frac{y_{(i)} \sum_{j \in S}jy_{(j)}}{n\sum_{j \in S}y_j}+\frac{\sum_{j \in S}jy_{(j)}}{n(n-1)} - \frac{\sum_{j \in S}y_j-\sum_{j=1}^{i}y_{(j)} +iy_{(i)} }{n-1} \right]-\frac{1}{n(n-1)},$ with $$i=\{1,\ldots,n\}$$ being the jackknife estimates, i.e., $$\widehat{G}_{-i}$$ is the estimation of the Gini index when the unit $$i$$ is removed from the sample. For a confidence level $$1-\alpha$$, the "zjackknife" confidence interval is defined as $\left[\widehat{G} - Z_{1-\alpha/2}\sqrt{\widehat{V}_{J}(\widehat{G})}, \widehat{G} + Z_{1-\alpha/2}\sqrt{\widehat{V}_{J}(\widehat{G})} \right],$ where $$Z_{1-\alpha/2}$$ is the $$(1-\alpha/2)$$th quantile of the standard Normal distribution.


# Gini index estimation and confidence interval using 'zjackknife'.

igini(y, interval = "zjackknife")
#> $Gini #> [1] 0.4671929 #> #>$Interval
#>          lower     upper
#> [1,] 0.4103563 0.5240296
#>
#> $Variance #> [1] 0.0008409313 "tjackknife" sustitutes the critical value $$Z_{1-\alpha/2}$$ by critical values computed from the studentized bootstrap. This confidence interval is given by $\left[\widehat{G} - t_{J;1-\alpha/2}^{*}\sqrt{\widehat{V}_{J}(\widehat{G})}, \widehat{G} - t_{J;\alpha/2}^{*}\sqrt{\widehat{V}_{J}(\widehat{G})} \right],$ where $$t_{J;a}^{*}$$ is the $$a$$th quantile of the values $t^{*}_{J}(b)=\frac{\widehat{G}^{*}(b) - \widehat{G}}{\sqrt{\widehat{V}_{J}\left[\widehat{G}^{*}(b)\right]}}$ computed using the bootstrap technique, where $$\widehat{V}_{J}\left[\widehat{G}^{*}(b)\right]$$ is the estimated Ogwang Jackknife variance of $$\widehat{G}^{*}(b)$$ for the $$b$$th bootstrap sample.  # Gini index estimation and confidence interval using 'tjackknife'. igini(y, interval = "tjackknife") #>$Gini
#> [1] 0.4671929
#>
#> $Interval #> lower upper #> [1,] 0.4108575 0.5349601 #> #>$Variance
#> [1] 0.0008409313

### 1.2.3 Linearization

The linearization technique for variance estimation (Deville, 1999) has been applied to the following estimators of the Gini index: $\widehat{G}^{a} = \displaystyle \frac{1}{2\overline{y}n^{2}}\sum_{i \in S}\sum_{j\in S} |y_i-y_j|$ and $\widehat{G}^{b} = \displaystyle \frac{2}{\overline{y}n}\sum_{i \in S}y_{i}\widehat{F}_{n}(y_{i}) - 1,$ where $\widehat{F}_{n}(y_i)=\frac{1}{n}\sum_{j \in S}\delta(y_j \leq y_i)$ and $$\delta(\cdot)$$ is the indicator variable that takes the value 1 when its argument is true and 0 otherwise. For a given estimator $$\widehat{G}$$ and a linearizated variable $$z$$, the confidence interval, with confidence level $$1-\alpha$$, is defined as:
$\left[\widehat{G} - Z_{1-\alpha/2}\sqrt{\widehat{V}_{L}(\widehat{G})}, \widehat{G} + Z_{1-\alpha/2}\sqrt{\widehat{V}_{L}(\widehat{G})} \right],$

where the variance estimator of the Gini index is given by: $\widehat{V}_{L}(\widehat{G})= \displaystyle \frac{1}{n(n-1)}\sum_{i \in S}\left(z_{i} - \overline{z}\right)^2,$ and $\overline{z}=\frac{1}{n}\sum_{i \in S}z_{i}.$

On the one hand, interval = "zalinearization" linearizates the estimator $$\widehat{G}^{a}$$, and the corresponding pseudo-values are (see Langel anf Tillé 2013):

$z_{(i)}^{a}=\frac{1}{\overline{y}}\left[ \frac{2i}{n}\left( y_{(i)} - \widehat{\overline{Y}}_{(i)}\right) + \overline{y} - y_{(i)} - \widehat{G}^{a}\left(\overline{y} + y_{(i)}\right) \right],$ where $\widehat{\overline{Y}}_{(i)} = \displaystyle \frac{1}{i}\sum_{j = 1}^{i}y_{(j)}.$

On the other hand, interval = "zblinearization" linearizates the estimator $$\widehat{G}^{b}$$, and the corresponding pseudo values are (see Berger, 2008):

$z_i^{b}=\frac{1}{\overline{y}}\left[ 2y_i\widehat{F}_{n}(y_i) - (\widehat{G}^{b}+1)(y_i+\overline{y})+2\frac{\sum_{j \in S}y_j\delta(y_j \geq y_i)}{n} \right].$


# Gini index estimation and confidence interval using 'zalinearization'.

igini(y, interval = "zalinearization")
#> $Gini #> [1] 0.4671929 #> #>$Interval
#>          lower     upper
#> [1,] 0.4125876 0.5217982
#>
#> $Variance #> [1] 0.0007762 # Gini index estimation and confidence interval using 'zblinearization'. igini(y, interval = "zblinearization") #>$Gini
#> [1] 0.4671929
#>
#> $Interval #> lower upper #> [1,] 0.4107537 0.5236321 #> #>$Variance
#> [1] 0.0008292117

Intervals "talinearization" and "tblinearization" substitute the critical value $$Z_{1-\alpha/2}$$ by critical values computed from the Studentized bootstrap. This confidence interval is given by
$\left[\widehat{G} - t_{L;1-\alpha/2}^{*}\sqrt{\widehat{V}_{L}(\widehat{G})}, \widehat{G} - t_{L;\alpha/2}^{*}\sqrt{\widehat{V}_{L}(\widehat{G})} \right],$ where $$t_{L;a}^{*}$$ is the $$a$$th quantile of the values $t^{*}_{L}(b)=\frac{\widehat{G}^{*}(b) - \widehat{G}}{\sqrt{\widehat{V}_{L}\left[\widehat{G}^{*}(b)\right]}}.$ $$\widehat{V}_{L}(\cdot)$$ is computed using the pseudo-values $$z_{(i)}^{a}$$ when interval = "zalinearization", and using the pseudo-values $$z_i^{b}$$ when interval = "zblinearization".


# Gini index estimation and confidence interval using 'talinearization'.

igini(y, interval = "talinearization")
#> $Gini #> [1] 0.4671929 #> #>$Interval
#>          lower     upper
#> [1,] 0.4195142 0.5253662
#>
#> $Variance #> [1] 0.0007762 # Gini index estimation and confidence interval using 'tblinearization'. igini(y, interval = "tblinearization") #>$Gini
#> [1] 0.4671929
#>
#> $Interval #> lower upper #> [1,] 0.4224278 0.5329734 #> #>$Variance
#> [1] 0.0008292117

### 1.2.4 Empirical likelihood

Intervals "ELchisq" and "ELboot" compute the empirical likelihood ($$EL$$) method, a nonparametric technique that provides desirable inferences under skewed distributions. The shape of the $$EL$$ confidence intervals are determined by the data-driven likelihood ratio function (Owen, 2001). interval = "ELchisq" obtains the $$EL$$ confidence interval, with confidence level $$1-\alpha$$, for the Gini index as defined by Qin et al. (2010): $\left\{ \theta|-2R(\theta) \leq \frac{\chi^2_{1;1-\alpha}}{k}\right\}$

where $R(\theta)= - \sum_{i \in S} log\{1+\lambda Z(y_i,\theta)\}$ is the log-EL ratio statistic for $$\theta = G$$, $Z(y_i,\theta)=\{2\widehat{F}_{n}(y_i)-1\}y_{i} - \theta y_i,$ $$\lambda$$ is the solution to $\frac{1}{n}\sum_{i \in S}\frac{Z(y_i,\theta)}{1+Z(y_i,\theta)}=0,$ $$k=\widehat{\sigma}_{2}^{2}/\widehat{\sigma}_{1}^{2}$$ is the scaling factor, $\widehat{\sigma}_{j}^{2}=\frac{1}{n-1}\sum_{i \in S}\left(u_{ji} - \overline{u}_{j} \right)^2,$ with $$j=\{1,2\}$$, $\overline{u}_{j} = \frac{1}{n}\sum_{i \in S}u_{ji},$ and $$\chi^2_{1;1-\alpha}$$ is the $$(1-\alpha)$$th quantile of Chi-Squared distribution with one degree of freedom.


# Gini index estimation and confidence interval using 'ELchisq'.

igini(y, interval = "ELchisq")
#> $Gini #> [1] 0.4671929 #> #>$Interval
#>          lower     upper
#> [1,] 0.4216374 0.5319404
#>
#> $Variance #> [1] 0.0008292117 interval = "ELboot" substitutes the critical value based on the Chi-Squared distribution by an empirical critical value based on bootstrap. "ELboot" computes the $$EL$$ confidence interval (Qin et al., 2010): $\left\{ \theta|-2R(\theta) \leq C_{1-\alpha}\right\},$ where $$C_{1-\alpha}$$ is the $$(1-\alpha)$$th quantile of the values $$\{-R_{1}^{*}(\widehat{G}),\ldots, -R_{B}^{*}(\widehat{G})\}$$, and where $$R_{b}^{*}(\widehat{G})$$ denotes the value of $$R(\theta)$$ computed from the $$b$$th bootstrap sample.  # Gini index estimation and confidence interval using 'ELboot'. igini(y, interval = "ELboot") #>$Gini
#> [1] 0.4671929
#>
#> $Interval #> lower upper #> [1,] 0.4118394 0.5413343 #> #>$Variance
#> [1] 0.0008343201

The function icompareCI() compares the various confidence intervals for the scenario of a sample derived from an infinite population. The argument plotCI = TRUE plots the results derived from the various available methods for constructing confidence intervals.


# Comparisons of variance estimators and confidence intervals.

icompareCI(y, plotCI = FALSE)
#>           interval    bc gini lowerlimit upperlimit var.gini
#> 1       zjackknife FALSE 0.46       0.41       0.52    8e-04
#> 2       zjackknife  TRUE 0.47       0.41       0.52    8e-04
#> 3       tjackknife FALSE 0.46       0.41       0.53    8e-04
#> 4       tjackknife  TRUE 0.47       0.42       0.54    8e-04
#> 5  zalinearization FALSE 0.46       0.41       0.52    8e-04
#> 6  zalinearization  TRUE 0.47       0.41       0.52    8e-04
#> 7  talinearization FALSE 0.46       0.41       0.52    8e-04
#> 8  talinearization  TRUE 0.47       0.42       0.52    8e-04
#> 9  zblinearization FALSE 0.46       0.41       0.52    8e-04
#> 10 zblinearization  TRUE 0.47       0.41       0.52    8e-04
#> 11 tblinearization FALSE 0.46       0.42       0.53    8e-04
#> 12 tblinearization  TRUE 0.47       0.42       0.53    8e-04
#> 13      pbootstrap FALSE 0.46       0.40       0.51    8e-04
#> 14      pbootstrap  TRUE 0.47       0.40       0.51    8e-04
#> 15             BCa FALSE 0.46       0.42       0.53    7e-04
#> 16             BCa  TRUE 0.47       0.42       0.53    8e-04
#> 17         ELchisq FALSE 0.46       0.42       0.53    8e-04
#> 18         ELchisq  TRUE 0.47       0.42       0.53    8e-04
#> 19          ELboot FALSE 0.46       0.41       0.53    7e-04
#> 20          ELboot  TRUE 0.47       0.42       0.54    7e-04

# 2 Finite populations

## 2.1 Estimators of the Gini index

For a finite population $$U$$, $$\{Y_{i}: i\in U\}$$ denotes a sequence, with size $$N$$, of nonnegative random variables with the same distribution as the variable of interest $$Y$$, and $$\{y_{i}: i\in U\}$$ are the population values of the variable of interest. A sample $$S$$ is selected from $$U$$ by using a sampling design with survey weights $$w_i$$, with $$i\in S$$. For example, the survey weights can be $$w_i = \pi_{i}^{-1}$$, where $$\pi_{i}=P(i\in S)$$ are the inclusion probabilities (Muñoz et al., 2023). The Gini index ($$G$$) is estimated using the observations of individuals selected in the sample $$\{y_{i}: i\in S\}$$, and the corresponding survey weights $$\{w_{i}: i\in S\}$$. The different methods for estimating the Gini index are (see also Muñoz et al., 2023):

• method = 1 (Langel and Tillé, 2013).

$\widehat{G}_{w1}= \displaystyle \frac{1}{2\widehat{N}^{2}\overline{y}_{w}}\sum_{i \in S}\sum_{j \in S}w_{i}w_{j}|y_{i}-y_{j}|,$ where $$\widehat{N}=\sum_{i \in S}w_i$$ and $\overline{y}_{w}= \frac{1}{\widehat{N}}\sum_{i \in S}w_{i}y_{i}.$

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

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

• method = 3 (Berger, 2008).

$\widehat{G}_{w3} = \displaystyle \frac{2}{\widehat{N}\overline{y}_{w}}\sum_{i \in S}w_{i}y_{i}\widehat{F}_{w}^{\ast}(y_{i})-1,$ where $\widehat{F}_{w}^{\ast}(t) = \displaystyle \frac{1}{\widehat{N}}\sum_{i \in S}w_{i}[\delta(y_i < t) + 0.5\delta(y_i = t)]$ is the smooth (mid-point) distribution function.

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

$\widehat{G}_{w4} = 1 - \displaystyle \frac{\overline{v}_{w}}{\overline{y}_{w}},$ where $$\overline{v}_{w}=\widehat{N}^{-1}\sum_{i \in S}w_{i}v_{i}$$ and $v_{i} = \displaystyle \frac{1}{\widehat{N} - w_{i}}\sum_{ \substack{j \in S\\ j\neq i}}\min(y_{i},y_{j}).$

• method = 5 (Lerman and Yitzhaki, 1989).

$\widehat{G}_{w5} = \displaystyle \frac{2}{\widehat{N}\overline{y}_{w}} \sum_{i \in S} w_{(i)}^{+}[y_{(i)} - \overline{y}_{w}]\left[ \widehat{F}_{w}^{LY}(y_{(i)}) - \overline{F}_{w}^{LY} \right],$ where $\widehat{F}_{w}^{LY}(y_{(i)}) = \displaystyle \frac{1}{\widehat{N}}\left(\widehat{N}_{(i-1)} + \frac{w_{(i)}^{+}}{2} \right)$ and $\overline{F}_{w}^{LY}=\frac{1}{\widehat{N}}\sum_{i \in S}w_{(i)}^{+}\widehat{F}_{w}^{LY}(y_{(i)}).$

In the finite population example, income and weights from the 2006 Austrian EU-SILC data set (laeken package) are used to estimate the Gini index in the Austrian region of Burgenland. The Gini index is estimated using fgini() and method = 2 (the default method).

data(eusilc, package="laeken")
y <- eusilc$eqIncome[eusilc$db040 == "Burgenland"]
w <- eusilc$rb050[eusilc$db040 == "Burgenland"]
fgini(y, w)
#> [1] 0.3205489

fginindex() can be used to estimate the Gini index using various estimation methods and both R and C++ codes. Efficiency comparisons between both implementations and with other functions available in other packages, such as laeken, DescTools, ineq or REAT, can be made using, for example, the function microbenchmark():


#Comparing the computation time for the various estimation methods and using R
microbenchmark::microbenchmark(
fginindex(y, w, method = 1,  useRcpp = FALSE),
fginindex(y, w, method = 2,  useRcpp = FALSE),
fginindex(y, w, method = 3,  useRcpp = FALSE),
fginindex(y, w, method = 4,  useRcpp = FALSE),
fginindex(y, w, method = 5,  useRcpp = FALSE)
)
#> Unit: microseconds
#>                                          expr    min      lq      mean   median
#>  fginindex(y, w, method = 1, useRcpp = FALSE) 1315.7 1422.95  2091.698  1564.65
#>  fginindex(y, w, method = 2, useRcpp = FALSE)   47.7   68.25   112.532   105.25
#>  fginindex(y, w, method = 3, useRcpp = FALSE) 3665.1 4164.85  6859.617  5122.20
#>  fginindex(y, w, method = 4, useRcpp = FALSE) 7737.1 9392.00 12269.216 12195.65
#>  fginindex(y, w, method = 5, useRcpp = FALSE)   60.9  109.25   190.398   143.90
#>        uq     max neval
#>   2003.60  7574.2   100
#>    133.25   282.4   100
#>   7488.70 31897.5   100
#>  14806.60 21488.4   100
#>    164.65  4365.5   100

# Comparing the computation time for the various estimation methods and using Rcpp
microbenchmark::microbenchmark(
fginindex(y, w, method = 1),
fginindex(y, w, method = 2),
fginindex(y, w, method = 3),
fginindex(y, w, method = 4),
fginindex(y, w, method = 5)
)
#> Unit: microseconds
#>                         expr   min     lq    mean median    uq   max neval
#>  fginindex(y, w, method = 1) 367.6 369.35 399.065 379.60 395.1 758.8   100
#>  fginindex(y, w, method = 2)  42.2  53.60  77.953  58.60  83.9 278.8   100
#>  fginindex(y, w, method = 3) 413.5 415.45 488.974 428.60 481.0 959.5   100
#>  fginindex(y, w, method = 4) 349.7 352.15 377.131 362.05 368.6 549.9   100
#>  fginindex(y, w, method = 5)  44.6  56.60  81.559  62.80  91.7 258.4   100

# Comparing the computation time for estimates of the Gini index in various R packages.

# Comparing 'method = 2', used also by the laeken package.

microbenchmark::microbenchmark(
fgini(y,w),
laeken::gini(y,w)
)
#> Unit: microseconds
#>                expr  min    lq   mean median    uq   max neval
#>         fgini(y, w) 28.6 32.25 40.654  35.45 41.55 132.4   100
#>  laeken::gini(y, w) 46.4 54.70 64.262  59.00 65.75 241.4   100

# Comparing 'method = 5', used also by the DescTools and REAT packages.

microbenchmark::microbenchmark(
fgini(y,w, method = 5),
DescTools::Gini(y,w),
REAT::gini(y, weighting = w)
)
#> Unit: microseconds
#>                          expr   min     lq    mean median     uq   max neval
#>       fgini(y, w, method = 5)  31.3  37.95  53.580  43.45  57.15 204.1   100
#>         DescTools::Gini(y, w)  79.4  96.00 120.604 103.85 121.75 669.8   100
#>  REAT::gini(y, weighting = w) 184.5 242.55 296.965 261.90 308.05 817.2   100

## 2.2 Variance estimation and confidence intervals

Jackknife and linearization tecniques compute pseudo-values (named as $$z_{i}$$, with $$i \in S$$) that require the use of an expression for the variance estimation. The function fgini() can compute the following type variance estimators using the argument varformula:

1. The Horvitz-Thompson ("HT") type variance estimator (Hortvitz and Thompson, 1952).

$\widehat{V}_{HT}(\widehat{G}_{w}) = \displaystyle \sum_{i\in S}\sum_{j\in S}\breve{\Delta}_{ij}w_{i}w_{j}z_{i}z_{j},$ which is computed when varformula = "HT", where $\breve{\Delta}_{ij}=\displaystyle \frac{\pi_{ij}-\pi_{i}\pi_{j}}{\pi_{ij}}.$

1. The Sen-Yates-Grundy ("SYG") type variance estimator (Sen, 1953; Yates and Grundy, 1953).

$\widehat{V}_{SYG}(\widehat{G}_{w}) = - \displaystyle \frac{1}{2}\sum_{i\in S}\sum_{j\in S}\breve{\Delta}_{ij}(w_{i}z_i-w_{j}z_{j})^{2},$ which is computed when varformula = "SYG".

1. The Hartley-Rao ("HR") type variance estimator (Hartley and Rao, 1962).

$\widehat{V}_{HR}(\widehat{G}_{w}) = \displaystyle \frac{1}{n-1}\sum_{i\in S}\sum_{\substack{j \in S\\ j < i}}\left(1-\pi_i-\pi_j + \frac{1}{n}\sum_{k\in U}\pi_{k}^{2} \right)(w_{i}z_i-w_{j}z_{j})^{2},$ which is computed when varformula = "HR".

Note that the "HT" variance estimator may give negative values, and the "SYG" variance estimator is suitable for fixed-size sampling designs. This implies that "SYG" should not be used under Poisson sampling. Fortunately, "HT" always give positive values under this sampling design. We observe that both Horvitz-Thompson and Sen-Yates-Grundy variance estimators depend on second (joint) inclusion probabilities (argument Pij). The Hàjek (1964) approximation $\pi_{ij}\cong \pi_{i}\pi_{j}\left[1- \displaystyle \frac{(1-\pi_{i})(1-\pi_{j})}{\sum_{i \in S}(1-\pi_{i})} \right]$ is used when the second (joint) inclusion probabilities are not available (Pij = NULL). Note that the Hàjek approximation is suggested for large-entropy sampling designs, large samples, and large populations (see Tille 2006; Berger and Tille, 2009; Haziza et al., 2008; Berger, 2011). For instance, this approximation is not recomended for highly-stratified samples (Berger, 2005). The Hartley-Rao variance estimator requires the first inclusion probabilities at the population level (argument PiU).

### 2.2.1 Bootstrap

For complex sampling designs, the rescaled bootstrap (Rao el al., 1992; Rust and Rao, 1996) can be used for variance estimation and construction of confidence intervals. interval = "pbootstrap" returns the confidence interval for the Gini index using the rescaled bootstrap with confidence limits obtained by the percentile method. For a given estimator $$\widehat{G}_{w}$$ and a confidence level $$1-\alpha$$, this confidence interval is given by $\left[ \widehat{G}^{*}_{w;\alpha/2}, \widehat{G}^{*}_{w;1-\alpha/2} \right],$

where $$\widehat{G}^{*}_{w;a}$$ is the $$a$$th quantile of the bootstrapped coefficients $$\widehat{G}^{*}_{w}(b)$$, with $$b=\{1,\ldots,B\}$$, and which are obtained by using the expression $$\widehat{G}_{w}$$ after substituting the original survey weights $$w_{i}$$ by the bootstrap weights $w_{i}^{*}(b)=w_{i}\frac{r_{i}n}{n-1},$ where $$r_{i}$$ is the number of times that $$i$$yh unit is selected by the bootstrap procedure. A variance estimator of the Gini index based on the rescaled bootstrap is defined as: $\widehat{V}_{B}(\widehat{G}_{w})= \displaystyle \frac{1}{B-1}\sum_{b=1}^{B}\left(\widehat{G}^{*}_{w}(b) - \overline{G}^{*}_{w} \right)^2,$ where $\overline{G}^{*}_{w}=\frac{1}{B}\sum_{b=1}^{B}\widehat{G}^{*}_{w}(b).$


# Gini index estimation and confidence interval using 'pbootstrap'.

fgini(y, w, interval = "pbootstrap")
#> $Gini #> [1] 0.3205489 #> #>$Interval
#>          lower     upper
#> [1,] 0.2935952 0.3453333
#>
#> $Variance #> [1] 0.0001664895 ### 2.2.2 Jackknife The "zjackknife" method computes the variance of the Gini index using the jackknife technique. For a given estimator $$\widehat{G}_{w}$$, the pseudo-values for variance estimation are defined as (see Berger, 2008): $z_{i}=\displaystyle \frac{1}{w_{i}}\left(1-\frac{w_{i}}{\widehat{N}}\right)\left(\widehat{G}_{w} - \widehat{G}_{w;-i}\right),$ where $$\widehat{G}_{w;-i}$$ denotes the estimator $$\widehat{G}_{w}$$ computed from $$S\setminus\{i\}$$, i.e., from the sample $$S$$ after removing the $$i$$th unit. For a confidence level $$1-\alpha$$, the "zjackknife" confidence interval is defined as $\left[\widehat{G}_{w} - Z_{1-\alpha/2}\sqrt{\widehat{V}(\widehat{G}_{w})}, \widehat{G}_{w} + Z_{1-\alpha/2}\sqrt{\widehat{V}(\widehat{G}_{w})} \right],$ where the variance $$\widehat{V}(\widehat{G}_{w})$$ is computed using the pseudo-values $$z_i$$ and any of the aforementioned type variance estimators (Horvitz-Thompson; Sen-Yates-Grundy; or Harley-Rao).  # Gini index estimation and confidence interval using 'zjackknife'. fgini(y, w, interval = "zjackknife") #>$Gini
#> [1] 0.3205489
#>
#> $Interval #> lower upper #> [1,] 0.2945728 0.346525 #> #>$Variance
#> [1] 0.0001756514

### 2.2.3 Linearization

The linearization technique for variance estimation (Deville, 1999) has been applied to the following estimators of the Gini index: $\widehat{G}_{w}^{a}= \displaystyle \frac{1}{2\widehat{N}^{2}\overline{y}_{w}}\sum_{i \in S}\sum_{j \in S}w_{i}w_{j}|y_{i}-y_{j}|,$

and $\widehat{G}_{w}^{b} = \displaystyle \frac{2}{\widehat{N}\overline{y}_{w}}\sum_{i \in S}w_{i}y_{i}\widehat{F}_{w}(y_{i})-1,$

where $\widehat{F}_{w}(t)=\frac{1}{\widehat{N}}\sum_{i \in S}w_i\delta(y_i \leq t)$ For a given estimator $$\widehat{G}_w$$ and a linearizated variable $$z$$, the confidence interval, with confidence level $$1-\alpha$$, is defined as
$\left[\widehat{G}_w - Z_{1-\alpha/2}\sqrt{\widehat{V}(\widehat{G}_w)}, \widehat{G}_w + Z_{1-\alpha/2}\sqrt{\widehat{V}(\widehat{G}_w)} \right],$

where the variance $$\widehat{V}(\widehat{G}_w)$$ is computed using the corresponding pseudo-values and any of the aforementioned type variance estimators (Horvitz-Thompson; Sen-Yates-Grundy, or Harley-Rao).

On the one hand, interval = "zalinearization" linearizates the estimator $$\widehat{G}_{w}^{a}$$, and the corresponding pseudo-values are defined as (Langel anf Tillé 2013):

$z_{(i)}^{a}=\frac{1}{\widehat{N}^{2}\overline{y}_w}\left[ 2\widehat{N}_{(i)}\left( y_{(i)} - \widehat{\overline{Y}}_{(i)}\right) + \widehat{N}\left\{ \overline{y}_{w} - y_{(i)} - \widehat{G}_{w}^{a}\left(\overline{y}_{w} + y_{(i)} \right) \right\} \right],$ where $\widehat{\overline{Y}}_{(i)} = \displaystyle \frac{1}{\widehat{N}_{(i)}}\sum_{j=1}^{i}w_{(j)}^{+}y_{(j)}.$

On the other hand, interval = "zblinearization" linearizates the estimator $$\widehat{G}_{w}^{b}$$, and the corresponding pseudo values are (see Berger, 2008):

$z_i^{b}=\frac{1}{\hat{N}\overline{y}_{w}}\left[ 2y_i\widehat{F}_{w}(y_i) - (\widehat{G}_{w}^{b}+1)(y_i+\overline{y}_{w})+\frac{2}{\hat{N}}\sum_{j \in S}w_jy_j\delta(y_j \geq y_i) \right],$ where $\widehat{F}_{w}(t) = \displaystyle \frac{1}{\widehat{N}}\sum_{i \in S}w_{i}\delta(y_i \leq t).$


# Gini index estimation and confidence interval using:
## a: The method 2 for point estimation.
## b: The method 'zalinearization' for variance estimation.
## c: The Sen-Yates-Grundy type variance estimator.
## d: The Hàjek approximation for the joint inclusion probabilities.
fgini(y, w, interval = "zalinearization")
#> $Gini #> [1] 0.3205489 #> #>$Interval
#>          lower    upper
#> [1,] 0.2946057 0.346492
#>
#> $Variance #> [1] 0.0001752056 # Gini index estimation and confidence interval using: ## a: The method 3 for point estimation. ## b: The method 'zblinearization' for variance estimation. ## c: The Sen-Yates-Grundy type variance estimator. ## d: The Hàjek approximation for the joint inclusion probabilities. fgini(y, w, method = 3, interval = "zblinearization") #>$Gini
#> [1] 0.3205489
#>
#> $Interval #> lower upper #> [1,] 0.2944802 0.3466175 #> #>$Variance
#> [1] 0.0001769051

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