1 Introduction

  1. Work tool for members of this research project.

  2. Mechanism to share data, R codes, results and conclusions of this research project

library(moments, warn.conflicts=FALSE)
library(stats4, warn.conflicts=FALSE)
library(splines, warn.conflicts=FALSE)
library(VGAM, warn.conflicts=FALSE)
## Warning: package 'VGAM' was built under R version 4.2.2

2 Research data

2.1 Probabilistic distributions: Pareto, Dagum, Lognormal, Gamma, Weibull

\label{fig:1} Empirical expected values of the coefficient of skewness ($\bar{\gamma}_{n}^{R}$) for different values of the Gini index ($G$). Various continuous probabilistic distributions are considered. Samples, with sizes $n=\{50,500\}$, are selected from infinite populations.

Empirical expected values of the coefficient of skewness (\(\bar{\gamma}_{n}^{R}\)) for different values of the Gini index (\(G\)). Various continuous probabilistic distributions are considered. Samples, with sizes \(n=\{50,500\}\), are selected from infinite populations.

3 The Gini index

3.1 Definitions in continous variables

We assume that \(Y\) is a nonnegative continuous random variable

  • Definition 1 (Lorenz, 1905; Langel and Tillé, 2013):

\[G = 2\left(0.5 - \int_{0}^{1} L_{\alpha} d\alpha \right),\] where: \[ L_{\alpha} = \frac{1}{\mu_{Y}}\int_{0}^{Y_{\alpha}}ydF_{Y}(y)=\frac{1}{\mu_{Y}}\int_{0}^{\alpha}Y_{u}du, \] \[\mu_{Y}=E[Y]=\int_{0}^{+\infty}yf(y)dy=\int_{0}^{+\infty}ydF_{Y}(y),\] \(\alpha \in [0,1]\), \(Y_{\alpha}=F_{Y}^{-1}(\alpha)\) is the \(\alpha\)-quantile of \(Y\) and \(F_{Y}^{-1}(\alpha)=\inf\{t:F_{Y}(t)\geq \alpha\}\) is the inverse of the distribution function.

  • Definition 2 (Kendall and Stuart, 1977):

\[G = \frac{D_{G}}{2 \mu_{Y}}= \frac{1}{2 \mu_{Y}} \int_{0}^{+\infty} \int_{0}^{+\infty} |x-y|dF_{Y}(x)dF_{Y}(y),\] where: \[D_{G}=E[|X-Y|]=\int_{0}^{+\infty} \int_{0}^{+\infty} |x-y|dF_{Y}(x)dF_{Y}(y).\] - Definition 3 (Qin et al., 2010; Berger and Gedik-Balay, 2020): \[G=\frac{1}{\mu_{Y}}\int_{0}^{+\infty}\{2F_{Y}(y)-1\}ydF_{Y}(y)=1 - \frac{1}{\mu_{Y}}\int_{0}^{+\infty}\{1-F_{Y}(y)\}^{2}dy.\]

  • Definition 4 (Anand, 1983; Lerman and Yitzhaki, 1984):

\[G=\frac{2}{\mu_{Y}}cov\{Y, F_{Y}(y)\}.\]

  • Definition 5 (Yitzhaki, 1998; Peng, 2011; Berger and Gedik-Balay, 2020):

\[G=1 - \frac{\mu_{Z}}{\mu_{Y}},\] where:

\[\mu_{Z} = E(Z)= \int_{0}^{+\infty}\{1-F_{Z}(z)\}dz\]

\(Z=\min\{Y_{a}, Y_{b}\}\), and \(Y_{a}\) and \(Y_{b}\) are two independent random variables with the same distribution as \(Y\).

4 OG1: The bias of the Gini index

5 OG2: Missing data and the Gini index

6 OG3: The GiniUs package in the statistical software R

7 OG4: Applications in Social Sciences and the social impact

8 OG5: Publications and internationalization of research results.

8.1 Working papers

  1. Exploring and correcting the bias in the estimation of the Gini measure of inequality. J.F.Muñoz, P.J.Moya and E.Álvarez.

  2. The problem of missing data in Strategic Management Research: critical issues, understanding and methodological insights. F.Vendrell, J.F.Muñoz, E.Gomes, D.W Lehman.

8.2 Published papers

  1. Álvarez, E., Moya, P.J. & Muñoz, J.F. (2021). Single imputation methods and confidence interval for the Gini index. Mathematics, 9, 3252.

8.3 International conferences of interest

8.4 Methodological and research seminars

10 References

Anand, S. (1983). Inequality and poverty in Malaysia: Measurement and decomposition. The World Bank.

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

Kendall, M., & Stuart, A. (1977). The advanced theory of statistics. Vol. 1: Distribution Theory. London: Griffin.

Langel, M., & 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., & Yitzhaki, S. (1984). A note on the calculation and interpretation of the Gini index. Economics Letters, 15(3-4), 363-368.

Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American Statistical Association, 9(70), 209-219.

Peng, L. (2011). Empirical likelihood methods for the Gini index. Australian & New Zealand Journal of Statistics, 53(2), 131-139.

Qin, Y., Rao, J. N. K., & Wu, C. (2010). Empirical likelihood confidence intervals for the Gini measure of income inequality. Economic Modelling, 27(6), 1429-1435.

Yitzhaki, S. (1998). More than a dozen alternative ways of spelling Gini. Research on Economic Inequality, 8, 13–30.