Package 'giniVarCI'

Comparisons of variance estimates and confidence intervals for the Gini index in finite populations

Description

Compares variance estimates and confidence intervals for the Gini index in finite populations.

Usage

fcompareCI(
  y,
  w,
  Pi = NULL,
  Pij = NULL,
  PiU,
  alpha = 0.05,
  B = 1000L,
  digitsgini = 2L,
  digitsvar = 4L,
  na.rm = TRUE,
  plotCI = TRUE,
  line.types = c(1L, 2L, 4L),
  colors = c("red", "green", "blue"),
  shapes = c(8L, 4L, 3L),
  save.plot = FALSE,
  large.sample = FALSE)

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, the variance estimation and the confidence interval. This argument can be missing if argument Pi is provided.
Pi	A numeric vector with the (sample) first inclusion probabilites to be used for estimating the Gini index, the variance estimation and the confidence interval. This argument can be NULL if argument w is provided. The default value is Pi = NULL.
Pij	A numeric square matrix with the (sample) second (joint) inclusion probabilites to be used for the variance estimation and the confidence interval. The Hajek approximation is used when Pij = NULL. This argument is used by the intervals "zjackknife", "zalinearization" and "zblinearization". The default value is Pij = NULL.
PiU	A numeric vector with the (population) first inclusion probabilites. The Hartley-Rao (HR) expression for the variance estimation is also computed if this argument is provided.
alpha	A single numeric value between 0 and 1 specifying the confidence level 1-alpha to be used for computing the confidence interval for the Gini index. Some authors call alpha the significance level. The default value is alpha = 0.05.
B	A single integer specifying the number of bootstrap replicates. The default value is B = 1000L.
digitsgini	A single integer specifying the number of decimals used in the estimation of the Gini index and confidence intervals. The default value is digitsgini = 2L.
digitsvar	A single integer specifying the number of decimals used in the variance estimation of the Gini index. The default value is digitsvar = 4L.
na.rm	A 'TRUE/FALSE' logical value indicating whether NA values should be removed before the computation proceeds. The default value is na.rm = TRUE.
plotCI	A 'TRUE/FALSE' logical value indicating whether confidence intervals are compared using a plot. The default value is plotCI = TRUE.
line.types	A numeric vector of length 3 specifying the line types. See the function plot for the different line types. The default value is line.types = c(1L, 2L, 4L).
colors	A vector of length 3 specifying the colors for lines of the plot. The default value is colors = c("red", "green", "blue").
shapes	A numeric vector specifying the point shapes for the limits of intervals. If PiU is missing, the function uses the two first components of shapes, i.e., it must have at least length 2. If PiU is provided, shapes must have at least length 3. See the function plot for the different point shapes. The default value is shapes = c(8L, 4L, 3L).
save.plot	A 'TRUE/FALSE' logical value indicating whether the ggplot object of the plot comparing the confidence intervals should be saved in the output. The default value is save.plot = FALSE.
large.sample	A 'TRUE/FALSE' logical value indicating whether the sample is large to apply a faster algorithm to sort the sample values in the computation of the Gini index. The default value is large.sample = FALSE.

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, variances and confidence intervals using various formulations. The different methods for estimating the Gini index are (see also Muñoz et al., 2023):

\ Gini Index formulae.

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 $\widehat{N}=\sum_{i \in S}w_i$, $\overline{y}_{w}=\widehat{N}^{-1}\sum_{i \in S}w_{i}y_{i}$, 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)

$\widehat{G}_{w2} =\displaystyle \frac{2\sum_{i \in S}w_{(i)}^{+}\widehat{N}_{(i)}y_{(i)} - \sum_{i \in S}w_{i}^{2}y_{i} }{\widehat{N}^{2}\overline{y}_{w}}-1,$

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 $\widehat{N}_{(i)}=\sum_{j=1}^{i}w_{(j)}^{+}$. Langel and Tillé (2013) show that $\widehat{G}_{w1} = \widehat{G}_{w2}$, so the computation of $\widehat{G}_{w1}$ is ommited in results.

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, and $\delta(\cdot)$ is the indicator variable that takes the value 1 when its argument is true, and 0 otherwise. It can be seen that $\widehat{G}_{w2} = \widehat{G}_{w3}$, so the computation of $\widehat{G}_{w3}$ is ommited in results.

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 $\overline{F}_{w}^{LY}=\widehat{N}^{-1}\sum_{i \in S}w_{(i)}^{+}\widehat{F}_{w}^{LY}(y_{(i)})$.

\ Variances and confidence intervals.

For a given estimator $\widehat{G}_{w}$ and variable $z$, the Horvitz-Thompson type variance estimator (Hortvitz and Thompson, 1952) is given by

$\widehat{V}_{HT}(\widehat{G}_{w}) = \displaystyle \sum_{i\in S}\sum_{j\in S}\breve{\Delta}_{ij}w_{i}w_{j}z_{i}z_{j},$

where

$\breve{\Delta}_{ij}=\displaystyle \frac{\pi_{ij}-\pi_{i}\pi_{j}}{\pi_{ij}}$

and $\pi_{ij}$ is the second (joint) inclusion probability of the individuals $i$ and $j$, i.e., $\pi_{ij}=P\{(i,j)\in S)\}$ (argument Pij).

The Sen-Yates-Grundy type variance estimator (Sen, 1953; Yates and Grundy, 1953) is defined as

$\widehat{V}_{SYG}(\widehat{G}_{w}) = - \displaystyle \frac{1}{2}\sum_{i\in S}\sum_{j\in S}\breve{\Delta}_{ij}(w_{i}z_i-w_{j}z_{j})^{2}$

.

The Hartley-Rao type variance estimator (Hartley and Rao, 1962) is given by

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

Note that the The Horvitz-Thompson variance estimator can give negative values. We observe that both Horvitz-Thompson and Sen-Yates-Grundy variance estimators depend on second (joint) inclusion probabilities (argument Pij). The Hajek (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 Hajek approximation is suggested for large-entropy sampling designs, large samples, and large populations (see Tille 2006; Berger and Tillé, 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). zjackknife computes the confidence interval based on the jackknife technique with critical values based on the Normal approximation. zalinearization and zblinearization compute the confidence intervals based on the linearization technique applied to the estimators

$\widehat{G}_{w}^{a} = \widehat{G}_{w1}$

and

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

respectively, where

$\widehat{F}_{w}(t)=\frac{1}{\widehat{N}}\sum_{i \in S}w_i\delta(y_i \leq t).$

Critical values are also based on the Normal approximation. pbootstrap computes the variance using the rescaled bootstrap, and the confidence interval is constructed using the percentile method. The vignette vignette("GiniVarInterval") contains a detailed description of the various methods for variance estimation and confidence intervals for the Gini index.

The following table summarises the various types of variances and confidence intervals that the function fcompareCI computes.

Interval	Variance	Critical values	References
_______________	______________	_________________	_________________________
zjackknife	Jackknife	Normal	Berger (2008)
zalinearization	Linearization	Normal	Langel and Tille (2013)
zblinearization	Linearization	Normal	Berger (2008)
pBootstrap	Rescaled bootstrap	Percentile bootstrap	Berger and Gedik-Balay (2020)

Value

If save.plot = FALSE, a data frame with columns:

1. interval. The method used to construct the confidence interval.

2. method. The method used to estimate the Gini index.

3. varformula. The type of formula for the variance estimator. Posible values are HT and SYG if argument PiU is missing, and HT, SYG amd HR if argument PiU is provided.

4. gini. The estimation of the Gini index.

5. lowerlimit. The lower limit of the confidence interval.

6. upperlimit. The upper limit of the confidence interval.

7. var.gini. The variance estimation for the estimator of the Gini index.

If save.plot = TRUE, a list with two components: (i) 'base.CI' a data frame of seven columns as just described and (ii) 'plot' a (ggplot) description of the plot, which is a list with components that contain the plot itself, the data, information about the scales, panels, etc. As a side-effect, a plot that compares the various methods for constructing confidence intervals for the Gini index is displayed. **ggplot2** is needed to be installed for this option to work.

If plotCI = TRUE, as a side-effect, a plot that compares the various methods for constructing confidence intervals for the Gini index is displayed. **ggplot2** is needed to be installed for this option to work.

Author(s)

Juan F Munoz [email protected]

Jose M Pavia [email protected]

Encarnacion Alvarez [email protected]

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. (2005). Variance estimation with highly stratified sampling designs with unequal probabilities. Australian & New Zealand Journal of Statistics, 47, 365–373.

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. (2011). Asymptotic consistency under large entropy sampling designs with unequal probabilities. Pakistan Journal of Statistics, 27, 407–426.

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

Berger, Y. G. and Tillé, Y. (2009). Sampling with unequal probabilities. In Sample Surveys: Design, Methods and Applications (eds. D. Pfeffermann and C. R. Rao), 39–54. Elsevier, Amsterdam.

Hajek, J. (1964). Asymptotic theory of rejective sampling with varying probabilities from a finite population. The Annals of Mathematical Statistics, 35, 4, 1491–1523.

Hartley, H. O., and Rao, J. N. K. (1962). Sampling with unequal probabilities and without replacement. The Annals of Mathematical Statistics, 350-374.

Haziza, D., Mecatti, F. and Rao, J. N. K. (2008). Evaluation of some approximate variance estimators under the Rao-Sampford unequal probability sampling design. Metron, LXVI, 91–108.

Horvitz, D. G. and Thompson, D. J. (1952). A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663–685.

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

Sen, A. R. (1953). On the estimate of the variance in sampling with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 5, 119–127.

Tillé, Y. (2006). Sampling Algorithms. Springer, New York.

Yates, F., and Grundy, P. M. (1953). Selection without replacement from within strata with probability proportional to size. Journal of the Royal Statistical Society B, 15, 253–261.

See Also

fgini, fginindex

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"]

# Estimation of the Gini index and confidence intervals using different methods.
fcompareCI(y, w)

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)
fcompareCI(y, w, plotCI = FALSE)

---

Gini index, variances and confidence intervals in finite populations

Description

Estimates the Gini index and computes variances and confidence intervals in finite populations.

Usage

fgini(
  y,
  w,
  method = 2L,
  interval = NULL,
  Pi = NULL,
  Pij = NULL,
  PiU,
  alpha = 0.05,
  B = 1000L,
  na.rm = TRUE,
  varformula = "SYG",
  large.sample = FALSE
)

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, the variance and the confidence interval. 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.
interval	A character string specifying the type of variance estimation and confidence interval to be used. Possible values are "zjackknife", "zalinearization", "zblinearization" and "pbootstrap". interval = NULL omits the computation of both variance and confidence interval. The default value is interval = NULL.
Pi	A numeric vector with the (sample) first inclusion probabilites to be used for estimating the Gini index, the variance and the confidence interval. This argument can be NULL if argument w is provided. The default value is Pi = NULL.
Pij	A numeric square matrix with the (sample) second (joint) inclusion probabilites to be used for the variance estimation and the confidence interval. The Hajek approximation is used when Pij = NULL. This argument is used when interval={"zjackknife", "zalinearization", "zblinearization"}. The default value is Pij = NULL.
PiU	A numeric vector with the (population) first inclusion probabilites. This argument is only required when the Hartley-Rao expression for the variance estimation is selected (varformula = "HR").
alpha	A single numeric value between 0 and 1. If interval is not NULL, the confidence level to be used for computing the confidence interval for the Gini is 1-alpha. Some authors call alpha the significance level. The default value is alpha = 0.05.
B	A single integer specifying the number of bootstrap replicates. This argument is required when interval = "pbootsptrap". The default value is B = 1000L.
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.
varformula	A character string specifying the type of formula to be used for the variance estimator when interval = {"zjackknife", "zalinearization", "zblinearization"}. Possible values are "HT" (Hortvitz-Thompson), "SYG" (Sen-Yates-Grundy) and "HR" (Hartley-Rao). The default value is varformula = "SYG".
large.sample	A 'TRUE/FALSE' logical value indicating indicating whether the sample is large to apply a faster algorithm to sort the sample values in the computation of the Gini index. The default value is large.sample = FALSE.

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. his function estimates the Gini index, variances and confidence intervals using various formulations. The different methods for estimating the Gini index are (see also Muñoz et al., 2023):

\ Gini Index formulae.

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 $\widehat{N}=\sum_{i \in S}w_i$, $\overline{y}_{w}=\widehat{N}^{-1}\sum_{i \in S}w_{i}y_{i}$, 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)

$\widehat{G}_{w2} =\displaystyle \frac{2\sum_{i \in S}w_{(i)}^{+}\widehat{N}_{(i)}y_{(i)} - \sum_{i \in S}w_{i}^{2}y_{i} }{\widehat{N}^{2}\overline{y}_{w}}-1,$

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 $\widehat{N}_{(i)}=\sum_{j=1}^{i}w_{(j)}^{+}$. 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, 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 $\widehat{G}_{w2} = \widehat{G}_{w3}$.

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 $\overline{F}_{w}^{LY}=\widehat{N}^{-1}\sum_{i \in S}w_{(i)}^{+}\widehat{F}_{w}^{LY}(y_{(i)})$.

\ Variances and confidence intervals.

For a given estimator $\widehat{G}_{w}$ and variable $z$, the Horvitz-Thompson type variance estimator (Hortvitz and Thompson, 1952)

$\widehat{V}_{HT}(\widehat{G}_{w}) = \displaystyle \sum_{i\in S}\sum_{j\in S}\breve{\Delta}_{ij}w_{i}w_{j}z_{i}z_{j}$

is computed when varformula = "HT", where

$\breve{\Delta}_{ij}=\displaystyle \frac{\pi_{ij}-\pi_{i}\pi_{j}}{\pi_{ij}}$

and $\pi_{ij}$ is the second (joint) inclusion probability of the individuals $i$ and $j$, i.e., $\pi_{ij}=P\{(i,j)\in S)\}$ (argument Pij).

The Sen-Yates-Grundy type variance estimator (Sen, 1953; Yates and Grundy, 1953)

$\widehat{V}_{SYG}(\widehat{G}_{w}) = - \displaystyle \frac{1}{2}\sum_{i\in S}\sum_{j\in S}\breve{\Delta}_{ij}(w_{i}z_i-w_{j}z_{j})^{2}$

is computed when varformula = "SYG", and the Hartley-Rao type variance estimator (Hartley and Rao, 1962)

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

is computed when varformula = "HR". Note that the The Horvitz-Thompson variance estimator can give negative values. We observe that both Horvitz-Thompson and Sen-Yates-Grundy variance estimators depend on second (joint) inclusion probabilities (argument Pij). The Hajek (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 Hajek 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). zjakknife computes the confidence interval based on the jackknife technique with critical values based on the Normal approximation. zalinearization and zblinearization compute the confidence intervals based on the linearization technique applied to the estimators

$\widehat{G}_{w}^{a} = \widehat{G}_{w1}$

and

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

respectively, where

$\widehat{F}_{w}(t)=\frac{1}{\widehat{N}}\sum_{i \in S}w_i\delta(y_i \leq t).$

Critical values are also based on the Normal approximation. pbootstrap computes the variance using the rescaled bootstrap, and the confidence interval is constructed using the percentile method. The vignette vignette("GiniVarInterval") contains a detailed description of the various methods for variance estimation and confidence intervals for the Gini index.

The following table summarises the various types of variances and confidence intervals that the function fgini computes. The argument varformula only applies for the jackknife and linearization techniques (see Berger, 2008; Langel and Tillé, 2013).

Interval	Variance	Critical values	References
_______________	______________	_________________	_________________________
zjackknife	Jackknife	Normal	Berger (2008)
zalinearization	Linearization	Normal	Langel and Tille (2013)
zblinearization	Linearization	Normal	Berger (2008)
pBootstrap	Rescaled bootstrap	Percentile bootstrap	Berger and Gedik-Balay (2020)

Value

When interval = NULL, the function returns a single numeric value between 0 and 1 informing about the estimation of the Gini index. When interval is not NULL, the function returns a list with 3 components: a single numeric value with the estimation of the Gini index; a single numeric value with the variance estimation of the Gini index; and a vector of length two containing the lower and upper limits of the confidence interval for the Gini index.

Author(s)

Juan F Munoz [email protected]

Jose M Pavia [email protected]

Encarnacion Alvarez [email protected]

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. (2005). Variance estimation with highly stratified sampling designs with unequal probabilities. Australian & New Zealand Journal of Statistics, 47, 365–373.

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. (2011). Asymptotic consistency under large entropy sampling designs with unequal probabilities. Pakistan Journal of Statistics, 27, 407–426.

Berger, Y. G. and Tillé, Y. (2009). Sampling with unequal probabilities. In Sample Surveys: Design, Methods and Applications (eds. D. Pfeffermann and C. R. Rao), 39–54. Elsevier, Amsterdam

Berger, Y., and Gedik-Balay, I. (2020). Confidence intervals of Gini coefficient under unequal probability sampling. Journal of Official Statistics, 36(2), 237-249.

Hajek, J. (1964). Asymptotic theory of rejective sampling with varying probabilities from a finite population. The Annals of Mathematical Statistics, 35, 4, 1491–1523.

Hartley, H. O., and Rao, J. N. K. (1962). Sampling with unequal probabilities and without replacement. The Annals of Mathematical Statistics, 350-374.

Haziza, D., Mecatti, F. and Rao, J. N. K. (2008). Evaluation of some approximate variance estimators under the Rao-Sampford unequal probability sampling design. Metron, LXVI, 91–108.

Horvitz, D. G. and Thompson, D. J. (1952). A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663–685.

Langel, M., and Tille, 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

Sen, A. R. (1953). On the estimate of the variance in sampling with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 5, 119–127.

Tillé, Y. (2006). Sampling Algorithms. Springer, New York.

Yates, F., and Grundy, P. M. (1953). Selection without replacement from within strata with probability proportional to size. Journal of the Royal Statistical Society B, 15, 253–261.

See Also

fginindex, 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"]

# Estimation of the Gini index using 'method = 2' .
fgini(y, w)


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)

# Gini index estimation and confidence interval using:
 ## a: The method 2 for point estimation.
 ## b: The method 'zjackknife' for variance estimation.
 ## c: The Sen-Yates-Grundy type variance estimator.
 ## d: The Hajek approximation for the joint inclusion probabilities.
fgini(y, w, interval = "zjackknife")

# 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 Hajek approximation for the joint inclusion probabilities.
fgini(y, w, interval = "zalinearization")

# 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 Hajek approximation for the joint inclusion probabilities.
fgini(y, w, method = 3L, interval = "zblinearization")

# Gini index estimation and confidence interval using:
 ## a: The method 2 for point estimation.
 ## b: The method 'pbootstrap' for variance estimation.
 ## c: The percentile bootstrap method for the confidence interval.
fgini(y, w, interval = "pbootstrap")

---

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
)

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.

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)

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

where $\widehat{N}=\sum_{i \in S}w_i$, $\overline{y}_{w}=\widehat{N}^{-1}\sum_{i \in S}w_{i}y_{i}$, 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)

$\widehat{G}_{w2} =\displaystyle \frac{2\sum_{i \in S}w_{(i)}^{*}\widehat{N}_{(i)}y_{(i)} - \sum_{i \in S}w_{i}^{2}y_{i} }{\widehat{N}^{2}\overline{y}_{w}}-1,$

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 $\widehat{N}_{(i)}=\sum_{j=1}^{i}w_{(j)}^{*}$. 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, 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 $\widehat{G}_{w2} = \widehat{G}_{w3}$.

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

$\widehat{G}_{w4} = 1 - \displaystyle \frac{\overline{z}_{w}}{\overline{y}_{w}},$

where $\overline{z}_{w}=\widehat{N}^{-1}\sum_{i \in S}w_{i}z_{i}$ and

$z_{i} = \displaystyle \frac{1}{\widehat{N} - w_{i}}\sum_{ \substack{j \in S\\ j\neq i}}\min(y_{i},y_{j}).$

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

$\widehat{F}_{w}^{LY}(y_{i}) = \displaystyle \frac{1}{\widehat{N}}\left(\widehat{N}_{(i-1)} + \frac{w_{(i)}^{\ast}}{2} \right)$

and $\overline{F}_{w}^{LY}=\widehat{N}^{-1}\sum_{i \in S}w_{i}\widehat{F}_{w}^{LY}(y_{i})$.

Value

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

Author(s)

Juan F Munoz [email protected]

Jose M Pavia [email protected]

Encarnacion Alvarez [email protected]

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)

---

Gini index for the Beta distribution with user-defined shape parameters

Description

Calculates the Gini index for the Beta distribution with shape parameters $a$ (shape1) and $b$ (shape2).

Usage

gbeta(shape1, shape2)

Arguments

shape1	A positive real number specifying the shape1 parameter $a$ of the Beta distribution.
shape2	A positive real number specifying the shape2 parameter $b$ of the Beta distribution.

Details

The Beta distribution with shape parameters $a$ (argument shape1) and $b$ (argument shape2) and denoted as $Beta(a,b)$, where $a>0$ and $b>0$, has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Yee, 2022)

$f(y) = \displaystyle \frac{1}{B(a,b)}y^{a-1}(1-y)^{b-1},$

and a cumulative distribution function given by

$F(y)= \displaystyle \frac{B(y;a,b)}{B(a,b)}$

where $0 \leq y \leq 1$,

$B(a,b) = \displaystyle \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$

is the beta function,

$\Gamma(\alpha) = \int_{0}^{\infty}t^{\alpha-1}e^{-t}dt$

is the gamma function, and

$B(y;a,b) = \displaystyle \int_{0}^{y}t^{a-1}(1-t)^{b-1}dt$

is the incomplete beta function.

The Gini index can be computed as

$G = \displaystyle \frac{2}{a}\frac{B(a+b,a+b)}{B(a,a)B(b,b)}.$

Value

A numeric value with the Gini index. A NA is returned when a shape parameter is non-numeric or non-positive.

Author(s)

Juan F Munoz [email protected]

Jose M Pavia [email protected]

Encarnacion Alvarez [email protected]

References

Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995). Continuous Univariate Distributions, volume 1, chapter 14. Wiley, New York.

Yee, T. W. (2022). VGAM: Vector Generalized Linear and Additive Models. R package version 1.1-7, https://CRAN.R-project.org/package=VGAM.

See Also

gf, gunif, gweibull, ggamma, gchisq

Examples

# Gini index for the Beta distribution with shape parameters 'a = 2' and 'b = 1'.
gbeta(shape1 = 2, shape2 = 1)

# Gini index for the Beta distribution with shape parameters 'a = 1' and 'b = 2'.
gbeta(shape1 = 1, shape2 = 2)

---

Gini index for the Burr Type XII (Singh-Maddala) distribution with user-defined scale and shape parameters

Description

Calculates the Gini index for the Burr Type XII (Singh-Maddala) distribution with scale parameter $b$ and shape parameters $g$ (shape.g) and $s$ (shape.s).

Usage

gburr(
 scale = 1,
 shape.g = 1,
 shape.s = 1
)

Arguments

scale	A positive real number specifying the scale parameter $b$ of the Burr Type XII (Singh-Maddala) distribution. The default value is scale = 1.
shape.g	A positive real number specifying the shape parameter $g$ of the Burr Type XII (Singh-Maddala) distribution. The default value is shape.g = 1.
shape.s	A positive real number specifying the shape parameter $s$ of the Burr Type XII (Singh-Maddala) distribution. The default value is shape.s = 1.

Details

The Burr Type XII (Singh-Maddala) distribution with scale parameter $b$, shape parameters $g$ (argument shape.g) and $s$ (argument shape.s) and denoted as $BurrXII(b,g,s)$, where $b>0$, $g>0$ and $s>0$, has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Rodriguez, 1977; Yee, 2022)

$f(y) = \displaystyle \frac{gs}{b}\left(\frac{y}{b}\right)^{g-1}\left[1 + \left(\frac{y}{b}\right)^{g}\right]^{-(s+1)},$

and a cumulative distribution function given by

$F(y)=1-\left[1 + \displaystyle \left( \frac{y}{b}\right)^{g} \right]^{-s},$

where $y>0$.

The Gini index can be computed as

$G = 2\left(0.5 - \displaystyle \frac{1}{E[y]}\int_{0}^{1}\int_{0}^{Q(y)}yf(y)dy\right),$

where $Q(y)$ is the quantile function of the Burr Type XII (Singh-Maddala) distribution, and $E[y]$ is the expectation of the distribution. The Burr Type XII (Singh-Maddala) distribution is related to the Pareto (IV) distribution: $BurrXII(b,g,s) = ParetoIV(0,b,1/g,s)$.

Value

A numeric value with the Gini index. A NA is returned when any of the parameter is non-numeric or non-positive.

Author(s)

Juan F Munoz [email protected]

Jose M Pavia [email protected]

Encarnacion Alvarez [email protected]

References

Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995). Continuous Univariate Distributions, volume 1, chapter 14. Wiley, New York.

Rodriguez, R. N. (1977). A guide to the Burr type XII distributions. Biometrika, 64(1), 129-134.

Yee, T. W. (2022). VGAM: Vector Generalized Linear and Additive Models. R package version 1.1-7, https://CRAN.R-project.org/package=VGAM.

See Also

gparetoIV, gpareto, gparetoI, gparetoII, gparetoIII, gfisk

Examples

# Gini index for the Burr Type XII distribution with 'scale = 1', 'shape.g = 2', 'shape.s = 1'.
gburr(scale = 1, shape.g = 2, shape.s = 1)

# Gini index for the Burr Type XII distribution with 'scale = 1', 'shape.g = 5', 'shape.s = 3'.
gburr(scale = 1, shape.g = 5, shape.s = 3)

---

Gini index for the Chi-Squared distribution with user-defined degrees of freedom

Description

Calculates Gini indices for the Chi-Squared distribution with degrees of freedom $n$ (df).

Usage

gchisq(df)

Arguments

df	A vector of positive real numbers specifying degrees of freedom of the Chi-Squared distribution.

Details

The Chi-Squared distribution with degrees of freedom $n$ (argument df) and denoted as $\chi_{n}^2$, where $n>0$, has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995)

$f(y)= \displaystyle \frac{1}{2^{n/2}\Gamma\left(\frac{n}{2}\right)}y^{n/2-1}e^{-y/2},$

and a cumulative distribution function given by

$F(y) = \frac{\gamma\left(\frac{n}{2}, \frac{y}{2}\right)}{\Gamma(\alpha)},$

where $y \geq 0$, the gamma function is defined by

$\Gamma(\alpha) = \int_{0}^{\infty}t^{\alpha-1}e^{-t}dt,$

and the lower incomplete gamma function is given by

$\gamma(\alpha,y) = \int_{0}^{y}t^{\alpha-1}e^{-t}dt.$

The Gini index can be computed as

$G=\displaystyle \frac{2\Gamma\left( \frac{1+n}{2}\right)}{n\Gamma\left(\frac{n}{2}\right)\sqrt{\pi}}.$

The Chi-Squared distribution is related to the Gamma distribution: $\chi_{n}^2 = Gamma(n/2, 2)$.

Value

A numeric vector with the Gini indices. A NA is returned when degrees of freedom are non-numeric or non-positive.

Author(s)

Juan F Munoz [email protected]

Jose M Pavia [email protected]

Encarnacion Alvarez [email protected]

References

Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1, chapter 14. Wiley, New York.

See Also

ggamma, gf, gbeta, glnorm

Examples

# Gini index for the Chi-Squared distribution with degrees of freedom equal to 2.
gchisq(df = 2)

# Gini indices for the Chi-Squared distribution and different degrees of freedom.
gchisq(df = 5:10)

---

Gini index for the Dagum distribution with user-defined shape parameters

Description

Calculates the Gini index for the Dagum distribution with shape parameters $a$ (shape1.a) and $p$ (shape2.p).

Usage

gdagum(shape1.a, shape2.p)

Arguments

shape1.a	A positive real number specifying the shape1 parameter $a$ of the Dagum distribution.
shape2.p	A positive real number specifying the shape parameter $p$ of the Dagum distribution.

Details

The Dagum distribution with scale parameter $b$, shape parameters $a$ (argument shape1.a) and $p$ (argument shape2.p) and denoted as $Dagum(b,a,p)$ , where $b>0$, $a>0$ and $p>0$, has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Rodriguez, 1977; Yee, 2022)

$f(y) = \displaystyle \frac{ap}{y}\frac{\left(\frac{y}{b}\right)^{ap}}{ \left[\left(\frac{y}{b} \right)^{a} + 1 \right]^{p+1} },$

and a cumulative distribution function given by

$F(y)= \left[1 + \displaystyle \left( \frac{y}{b}\right)^{-a} \right]^{-p},$

where $y > 0$.

The Gini index can be computed as

$G = \displaystyle \frac{\Gamma(p)\Gamma(2p+1/a)}{\Gamma(2p)\Gamma(p+1/a)}-1,$

where the gamma function is defined as

$\Gamma(\alpha) = \int_{0}^{\infty}t^{\alpha-1}e^{-t}dt.$

The Dagum distribution is also known the Burr III, inverse Burr, beta-K, or 3-parameter kappa distribution. The Dagum distribution is related to the Fisk (Log Logistic) distribution: $Dagum(b,a,1) = Fisk(b,a)$. The Dagum distribution is also related to the inverse Lomax distribution and the inverse paralogistic distribution (see Kleiber and Kotz, 2003; Johnson et al., 1995; Yee, 2022).

Value

A numeric value with the Gini index. A NA is returned when a shape parameter is non-numeric or non-positive.

Note

The Gini index of the Dagum distribution does not depend on its scale parameter.

Author(s)

Juan F Munoz [email protected]

Jose M Pavia [email protected]

Encarnacion Alvarez [email protected]

References

Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1, chapter 14. Wiley, New York.

Yee, T. W. (2022). VGAM: Vector Generalized Linear and Additive Models. R package version 1.1-7, https://CRAN.R-project.org/package=VGAM.

See Also

gburr, gpareto, gfisk, ggompertz, gfrechet

Examples

# Gini index for the Dagum distribution with shape parameters 'a = 2' and 'p = 20'.
gdagum(shape1.a = 2, shape2.p = 20)

---

Gini index for the F distribution with user-defined degrees of freedom

Description

Calculates the Gini index for the F distribution with degrees of freedom $\nu_1$ (df1) and $\nu_2$ (df2).

Usage

gf(df1, df2)

Arguments

df1	A positive real number specifying the degrees of freedom $\nu_1$ of the F distribution.
df2	A positive real number higher or equal than two specifying the degrees of freedom $\nu_2$ of the F distribution.

Details

The F distribution with $\nu_1$ (argument df1) and $\nu_2$ (argument df2) degrees of freedom and denoted as $F_{\nu_1,\nu_2}$, where $\nu_1>0$ and $\nu_2 > 0$, has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995)

$f(y) = \displaystyle \frac{\Gamma\left(\frac{\nu_{1}}{2} + \frac{\nu_{2}}{2}\right)}{\Gamma\left(\frac{\nu_{1}}{2}\right)\Gamma\left(\frac{\nu_{2}}{2}\right)}\left( \frac{\nu_{1}}{\nu_{2}}\right)^{\nu_{1}/2}y^{\nu_{1}/2-1}\left(1 + \frac{\nu_{1}y}{\nu_{2}}\right)^{-(\nu_{1}+\nu_{2})/2},$

and a cumulative distribution function given by

$F(y)= \displaystyle I_{\nu_{1}y/(\nu_{1}y + \nu_{2})}\left( \frac{\nu_{1}}{2}, \frac{\nu_{2}}{2} \right),$

where $y \geq 0$,

$\Gamma(\alpha) = \int_{0}^{\infty}t^{\alpha-1}e^{-t}dt$

is the gamma function,

$I_{y}(a,b)=\displaystyle \frac{B(y;a,b)}{B(a,b)}$

is the regularized incomplete beta function,

$B(a,b) = \displaystyle \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$

is the beta function, and

$B(y;a,b) = \displaystyle \int_{0}^{y}t^{a-1}(1-t)^{b-1}dt$

is the incomplete beta function.

The Gini index, for $\nu_2 \geq 2$, can be computed as

$G = 2\left(0.5 - \displaystyle \frac{\nu_{2} - 2}{ \nu_{2}}\int_{0}^{1}\int_{0}^{Q(y)}yf(y)dy\right),$

where $Q(y)$ is the quantile function of the F distribution.

Value

A numeric value with the Gini index. A NA is returned when degrees of freedom are non-numeric or $df1 \leq 0$ or $df2 < 2$ .

Author(s)

Juan F Munoz [email protected]

Jose M Pavia [email protected]

Encarnacion Alvarez [email protected]

References

Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1, chapter 14. Wiley, New York.

See Also

gchisq, ggamma, ggompertz, glnorm

Examples

# Gini index for the F distribution with 'df1 = 10' and 'df2 = 20' degrees of freedom.
gf(df1 = 10, df2 = 20)

---

Gini index for the Fisk (Log Logistic) distribution with user-defined shape parameters

Description

Calculates the Gini indices for the Fisk (Log Logistic) distribution with shape parameters $a$ (shape1.a).

Usage

gfisk(shape1.a)

Arguments

shape1.a

A vector of positive real numbers specifying shape parameters $a$ of the Fisk (Log Logistic) distribution.

Details

The Fisk (Log Logistic) distribution with scale parameter $b$, shape parameter $a$ (argument shape1.a) and denoted as $Fisk(b,a)$, where $b>0$ and $a>0$, has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Yee, 2022)

$f(y) = \displaystyle \frac{a}{y}\frac{\left(\frac{y}{b}\right)^{a}}{ \left[\left(\frac{y}{b} \right)^{a} + 1 \right]^{2} },$

and a cumulative distribution function given by

$F(y)=1-\left[1 + \displaystyle \left( \frac{y}{b}\right)^{a} \right]^{-1},$

where $y \geq 0$.

The Gini index can be computed as

$G = \left\{ \begin{array}{cl} 1 , & 0< a <1; \\ \displaystyle \frac{1}{a}, & a \geq 1. \end{array} \right.$

The Fisk (Log Logistic) distribution is related to the Dagum distribution: $Fisk(b,a) = Dagum(b,a,1)$.

Value

A numeric vector with the Gini indices. A NA is returned when a shape parameter is non-numeric or non-positive.

Note

The Gini index of the Fisk (Log Logistic) distribution does not depend on its scale parameter.

Author(s)

Juan F Munoz [email protected]

Jose M Pavia [email protected]

Encarnacion Alvarez [email protected]

References

Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1, chapter 14. Wiley, New York.

Yee, T. W. (2022). VGAM: Vector Generalized Linear and Additive Models. R package version 1.1-7, https://CRAN.R-project.org/package=VGAM.

See Also

gdagum, gburr, gpareto, ggompertz

Examples

# Gini index for the Fisk distribution with a shape parameter 'a = 2'.
gfisk(shape1.a = 2)

# Gini indices for the Fisk distribution and different shape parameters.
gfisk(shape1.a = 1:10)

---

Gini index for the Frechet distribution with user-defined shape parameters

Description

Calculates the Gini indices for the Frechet distribution with shape parameters $s$.

Usage

gfrechet(shape)

Arguments

shape

A vector of positive real numbers higher or equal than 1 specifying shape parameters $s$ of the Frechet distribution.

Details

The Frechet distribution with location parameter $a$, scale parameter $b$, shape parameter $s$ and denoted as $Frechet(a,b,s)$, where $a>0$, $b>0$ and $s>0$, has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995)

$f(y) = \displaystyle \frac{sb}{(y-a)^{2}} \left(\frac{b}{y-a}\right)^{s-1} \exp\left[- \displaystyle \left(\frac{b}{y-a}\right)^{s} \right],$

and a cumulative distribution function given by

$F(y)= \displaystyle \exp\left[- \displaystyle \left(\frac{b}{y-a}\right)^{s} \right],$

where $y > a$.

The Gini index, for $s \geq 1$, can be computed as

$G = 2^{1/s} -1.$

Value

A numeric vector with the Gini indices. A NA is returned when a shape parameter is non-numeric or smaller than 1.

Note

The Gini index of the Frechet distribution does not depend on its location and scale parameters and only is defined when its shape parameter is at least 1.

Author(s)

Juan F Munoz [email protected]

Jose M Pavia [email protected]

Encarnacion Alvarez [email protected]

References

Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1, chapter 14. Wiley, New York.

See Also

gdagum, gburr, gfisk, gpareto, ggompertz

Examples

# Gini index for the Frechet distribution with a shape parameter 's = 1'.
gfrechet(shape = 1)

# Gini indices for the Frechet distribution and different shape parameters.
gfrechet(shape = 1:10)

---

Gini index for the Gamma distribution with user-defined shape parameter

Description

Calculates the Gini indices for the Gamma distribution with shape parameters $\alpha$.

Usage

ggamma(shape)

Arguments

shape

A vector of positive real numbers specifying the shape parameters $\alpha$ of the Gamma distribution.

Details

The Gamma distribution with shape parameter $\alpha$, scale parameter $\sigma$ and denoted as $Gamma(\alpha, \sigma)$, where $\alpha>0$ and $\sigma>0$, has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995)

$f(y) = \displaystyle \frac{1}{\sigma^{\alpha}\Gamma(\alpha)}y^{\alpha-1}e^{-y/\sigma},$

and a cumulative distribution function given by

$F(y) = \frac{\gamma\left(\alpha, \frac{y}{\sigma}\right)}{\Gamma(\alpha)},$

where $y \geq 0$, the gamma function is defined by

$\Gamma(\alpha) = \int_{0}^{\infty}t^{\alpha-1}e^{-t}dt,$

and the lower incomplete gamma function is given by

$\gamma(\alpha,y) = \int_{0}^{y}t^{\alpha-1}e^{-t}dt.$

The Gini index can be computed as

$G = \displaystyle \frac{\Gamma\left(\frac{2\alpha+1}{2}\right)}{\alpha\Gamma(\alpha)\sqrt{\pi}}.$

The Gamma distribution is related to the Chi-squared distribution: $Gamma(n/2, 2) = \chi_{n}^2$.

Value

A numeric vector with the Gini indices. A NA is returned when a shape parameter is non-numeric or non-positive.

Note

The Gini index of the Gamma distribution does not depend on its scale parameter.

Author(s)

Juan F Munoz [email protected]

Jose M Pavia [email protected]

Encarnacion Alvarez [email protected]

References

Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1, chapter 14. Wiley, New York.

See Also

gchisq, gf, gbeta, gweibull, glnorm

Examples

# Gini index for the Gamma distribution with 'shape = 1'.
ggamma(shape = 1)

# Gini indices for the Gamma distribution and different shape parameters.
ggamma(shape = 1:10)

---

Gini index for the Gompertz distribution with user-defined scale and shape parameters

Description

Calculate the Gini index for the Gompertz distribution with scale parameter $\beta$ and shape parameter $\alpha$.

Usage

ggompertz(
 scale = 1,
 shape
)

Arguments

scale	A positive real number specifying the scale parameter $\beta$ of the Gompertz distribution. The default value is scale = 1.
shape	A positive real number specifying the shape parameter $\alpha$ of the Gompertz distribution.

Details

The Gompertz distribution with scale parameter $\beta$, shape parameter $\alpha$ and denoted as $Gompertz(\beta, \alpha)$, where $\beta>0$ and $\alpha>0$, has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Rodriguez, 1977; Yee, 2022)

$f(y)= \alpha e^{\beta y} \exp\left[ - \displaystyle \frac{\alpha}{\beta}\left(e^{\beta y} - 1 \right) \right],$

and a cumulative distribution function given by

$F(y)= 1 -\exp\left[ - \displaystyle \frac{\alpha}{\beta}\left(e^{\beta y} - 1 \right) \right],$

where $y \geq 0$.

The Gini index can be computed as

$G = 2\left(0.5 - \displaystyle \frac{1}{E[y]}\int_{0}^{1}\int_{0}^{Q(y)}yf(y)dy\right),$

where $Q(y)$ is the quantile function of the Gompertz distribution, and $E[y]$ is the expectation of the distribution. If scale is not specified it assumes the default value of 1.

Value

A numeric value with the Gini index. A NA is returned when a parameter is non-numeric or non-positive.

Author(s)

Juan F Munoz [email protected]

Jose M Pavia [email protected]

Encarnacion Alvarez [email protected]

References

Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1, chapter 14. Wiley, New York.

Yee, T. W. (2022). VGAM: Vector Generalized Linear and Additive Models. R package version 1.1-7, https://CRAN.R-project.org/package=VGAM.

See Also

ggamma, gbeta, gchisq, gpareto

Examples

# Gini index for the Gompertz distribution with 'scale = 1' and 'shape = 3'.
ggompertz(scale = 1, shape = 3)

---