# Lorenz Surfaces Based on the Sarmanov–Lee Distribution with Applications to Multidimensional Inequality in Well-Being

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary Results

#### 2.1. Univariate Lorenz and Concentration Curves

**Definition**

**1.**

#### 2.2. The Three Altermative Definitions of Bivariate Lorenz Surface

**Definition**

**2.**

**Definition**

**3.**

- The marginal LCs can be obtained as $L({u}_{1};{F}_{1})={\mathrm{lim}}_{{u}_{2}\to \infty}L({u}_{1},{u}_{2};{F}_{12})$ and $L({u}_{2};{F}_{2})={\mathrm{lim}}_{{u}_{1}\to \infty}L({u}_{1},{u}_{2};{F}_{12})$.
- The bivariate Lorenz surface does not depend on changes of scale in the marginals.
- If ${F}_{12}$ is a product distribution function, then$$L({u}_{1},{u}_{2};{F}_{12})=L({u}_{1};{F}_{1})L({u}_{2};{F}_{2}),$$
- In the case of a product distribution, the two-attribute Gini defined in (4) can be written as,$$1-G\left({F}_{12}\right)=[1-G\left({F}_{1}\right)][1-G\left({F}_{2}\right)].$$

**Definition**

**4.**

#### 2.3. Bivariate Lorenz Surface Based on Bivariate Beta-Generated Distributions

#### 2.4. Bivariate Generalized Gini Index

## 3. The Bivariate Sarmanov–Lee Lorenz Surface

#### 3.1. The Bivariate Sarmanov–Lee Distribution

#### 3.2. The Bivariate Sl Lorenz Surface

**Theorem**

**1.**

#### 3.3. Bivariate Generalized Gini Index

**Theorem**

**2.**

**Proof.**

## 4. Bivariate Lorenz Surface Models

#### 4.1. Bivariate Power Lorenz Surfaces Based on the Fgm Family

#### 4.2. Bivariate Sarmanov–Lee Lorenz Surfaces with Beta and Gb1 Marginals

#### 4.3. Bivariate Sl Lorenz Surfaces with Gamma Marginals

#### 4.4. Bivariate Sl Lorenz Surfaces with Lognormal Marginals

#### 4.5. Other Classes of Bivariate Lorenz Surfaces

## 5. Extensions to Higher Dimensions and Stochastic Dominance

## 6. Application: Multidimensional Inequality in Well-Being

#### 6.1. Data and Estimation Methods

- Moments estimation of the marginal distributions. We define,$${m}_{i}=\frac{1}{n}\sum _{j=1}^{n}{x}_{ij},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{s}_{i}^{2}=\frac{1}{n}\sum _{j=1}^{n}{({x}_{ij}-{m}_{i})}^{2},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}i=1,2$$$$\begin{array}{ccc}\hfill {\widehat{a}}_{i}& =& \frac{{m}_{i}({m}_{i}-{m}_{i}^{2}-{s}_{i}^{2})}{{s}_{i}^{2}},\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\widehat{b}}_{i}& =& \frac{(1-{m}_{i})({m}_{i}-{m}_{i}^{2}-{s}_{i}^{2})}{{s}_{i}^{2}}.\hfill \end{array}$$
- Moment estimation of the dependence parameter. The estimate of w is based on the simple relation $\rho =w{\sigma}_{1}{\sigma}_{2}$. Then, if r denotes the sample linear correlation coefficient, and ${s}_{i}$, $i=1,2$ the sample standard deviation of the marginal distributions ${X}_{i}$, $i=1,2$, the point estimate of w is,$$\widehat{w}=\frac{r}{{s}_{1}\xb7{s}_{2}}.$$

#### 6.2. Results

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

LC | Lorenz curve |

GB1 | Generalised beta of the first kind |

FGM | Farlie–Gumbel–Morgenstern |

SL | Sarmanov–Lee |

DHS | Demographic and Health Surveys |

UNDP | United Nations Development Program |

## References

- Sen, A. The Concept of Development. In Handbook of Development Economics; Chenery, H., Srinivasan, T.N., Eds.; Elsevier: Amsterdam, The Netherlands, 1988; pp. 9–26. [Google Scholar]
- Sen, A. Development as Capabilities Expansion. J. Dev. Plan.
**1989**, 19, 41–58. [Google Scholar] - Sen, A. Development as Freedom; Oxford University Press: Oxford, UK, 1999. [Google Scholar]
- Jorda, V.; López-Noval, B.; Sarabia, J.M. Distributional dynamics of life satisfaction in Europe. J. Happiness Stud.
**2019**, 20, 1015–1039. [Google Scholar] [CrossRef] [Green Version] - Jorda, V.; Sarabia, J.M. International convergence in well-being indicators. Soc. Indic. Res.
**2015**, 120, 1–27. [Google Scholar] [CrossRef] - Atkinson, A.B. Multidimensional deprivation: Contrasting social welfare and counting approaches. J. Econ. Inequal.
**2003**, 1, 51–65. [Google Scholar] [CrossRef] - Atkinson, A.B.; Bourguignon, F. The comparison of multi-dimensioned distributions of economic status. Rev. Econ. Stud.
**1982**, 49, 183–201. [Google Scholar] [CrossRef] - Decancq, K.; Lugo, M.A. Inequality of Wellbeing: A Multidimensional Approach. Economica
**2012**, 79, 721–746. [Google Scholar] [CrossRef] - Kolm, S.C. Multidimensional Equalitarianisms. Q. J. Econ.
**1977**, 91, 1–13. [Google Scholar] [CrossRef] - Maasoumi, E. The measurement and decomposition of multi-dimensional inequality. Econometrica
**1986**, 54, 991–997. [Google Scholar] [CrossRef] [Green Version] - Slottje, D.J. Relative Price Changes and Inequality in the Size Distribution of Various Components. J. Bus. Econ. Stat.
**1987**, 5, 19–26. [Google Scholar] - Tsui, K.Y. Multidimensional generalizations of the relative and absolute inequality indices: The Atkinson-Kolm-Sen approach. J. Econ. Theory
**1995**, 67, 251–265. [Google Scholar] [CrossRef] - Tsui, K.Y. Multidimensional inequality and multidimensional generalized entropy measures: An axiomatic derivation. Soc. Choice Welf.
**1999**, 16, 145–157. [Google Scholar] [CrossRef] [Green Version] - Taguchi, T. On the two-dimensional concentration surface and extensions of concentration coefficient and Pareto distribution to the two-dimensional case-I. Ann. Inst. Stat. Math.
**1972**, 24, 355–382. [Google Scholar] [CrossRef] - Taguchi, T. On the two-dimensional concentration surface and extensions of concentration coefficient and Pareto distribution to the two-dimensional case-II. Ann. Inst. Stat. Math.
**1972**, 24, 599–619. [Google Scholar] [CrossRef] - Arnold, B.C. Pareto Distributions; International Co-Operative Publishing House: Fairland, MD, USA, 1983. [Google Scholar]
- Koshevoy, G.; Mosler, K. The Lorenz zonoid of a multivariate distribution. J. Am. Stat. Assoc.
**1996**, 91, 873–882. [Google Scholar] [CrossRef] - Sarabia, J.M.; Jordá, V. Bivariate Lorenz Curves based on the Sarmanov–Lee Distribution. In Topics in Statistical Simulation, Springer Proceedings in Mathematics & Statistics; Springer: New York, NY, USA, 2014; Volume 114, pp. 447–455. [Google Scholar]
- Lee, M.-L.T. Properties of the Sarmanov Family of Bivariate Distributions. Commun. Stat. Theory Methods
**1996**, 25, 1207–1222. [Google Scholar] - Sarmanov, O.V. Generalized Normal Correlation and Two-Dimensional Frechet Classes. Doklady
**1966**, 168, 596–599. [Google Scholar] - Gastwirth, J.L. A general definition of the Lorenz curve. Econometrica
**1971**, 39, 1037–1039. [Google Scholar] [CrossRef] - Kakwani, N.C. Applications of Lorenz Curves in Economic Analysis. Econometrica
**1977**, 45, 719–728. [Google Scholar] [CrossRef] - Taguchi, T. On the structure of multivariate concentration - some relationships among the concentration surface and two variate mean difference and regressions. Comput. Stat. Data Anal.
**1988**, 6, 307–334. [Google Scholar] [CrossRef] - Arnold, B.C. Majorization and the Lorenz Curve; Lecture Notes in Statistics 43; Springer: New York, NY, USA, 1987. [Google Scholar]
- Koshevoy, G. Multivariate Lorenz majorization. Soc. Choice Welf.
**1995**, 12, 93–102. [Google Scholar] [CrossRef] - Koshevoy, G.; Mosler, K. Multivariate Gini indices. J. Multivar. Anal.
**1997**, 60, 252–276. [Google Scholar] [CrossRef] - Mosler, K. Multivariate Dispersion, Central Regions and Depth: The Lift Zonoid Approach; Springer: Berlin, Germany, 2002. [Google Scholar]
- Kleiber, C.; Kotz, S. Statistical Size Distributions in Economics and Actuarial Sciences; John Wiley: Hoboken, NJ, USA, 2003. [Google Scholar]
- Sarabia, J.M.; Prieto, F.; Jordá, V. Bivariate beta-generated distributions with applications to well-being data. J. Stat. Distrib. Appl.
**2014**, 1, 1–15. [Google Scholar] [CrossRef] [Green Version] - Alexander, C.; Cordeiro, G.M.; Ortega, E.M.M.; Sarabia, J.M. Generalized beta-generated distributions. Comput. Stat. Data Anal.
**2012**, 56, 1880–1897. [Google Scholar] [CrossRef] - Arnold, B.C.; Sarabia, J.M. Analytic Expressions for Multivariate Lorenz Surfaces. Sankhya A Indian J. Stat.
**2018**, 80, 84–111. [Google Scholar] [CrossRef] - Balakrishnan, N.; Lai, C.-D. Continuous Bivariate Distributions; Springer: New York, NY, USA, 2009. [Google Scholar]
- Donaldson, D.; Weymark, J.A. A single parameter generalization of the Gin index of inequality. J. Econ. Theory
**1980**, 22, 67–86. [Google Scholar] [CrossRef] - Kakwani, N.C. Income Inequality and Poverty, Methods and Estimation and Policy Applications; Oxford University Press: New York, NY, USA, 1980. [Google Scholar]
- Yitzhaki, S. On an extension of the Gini inequality index. Int. Econ. Rev.
**1983**, 24, 617–628. [Google Scholar] [CrossRef] - Huang, J.S.; Kotz, S. Modifications of the Farlie-Gumbel-Morgenstern distributions. A tough hill to climb. Metrika
**1999**, 49, 135–145. [Google Scholar] [CrossRef] - Bairamov, I.; Kotz, S. On a new family of positive quadrant dependent bivariate distributions. Int. Math. J.
**2003**, 3, 1247–1254. [Google Scholar] - Arnold, B.C.; Sarabia, J.M. Majorization and the Lorenz Order with Applications in Applied Mathematics and Economics; Springer: New York, NY, USA, 2018. [Google Scholar]
- McDonald, J.B. Some generalized functions for the size distribution of income. Econometrica
**1984**, 52, 647–663. [Google Scholar] [CrossRef] - Sarabia, J.M. Parametric Lorenz Curves: Models and Applications. In Modeling Income Distributions and Lorenz Curves; Chotikapanich, D., Ed.; Springer: New York, NY, USA, 2008; pp. 167–190. [Google Scholar]
- Sarabia, J.M.; Castillo, E.; Pascual, M.; Sarabia, M. Bivariate Income Distributions with Lognormal Conditionals. J. Econ. Inequal.
**2007**, 5, 371–383. [Google Scholar] [CrossRef] [Green Version] - Sarabia, J.M.; Castillo, E.; Slottje, D. An Ordered Family of Lorenz Curves. J. Econ.
**1999**, 91, 43–60. [Google Scholar] [CrossRef] - Sarabia, J.M.; Castillo, E.; Pascual, M.; Sarabia, M. Mixture Lorenz Curves. Econ. Lett.
**2005**, 89, 89–94. [Google Scholar] [CrossRef] - Hlasny, V. Nonresponse Bias in Inequality Measurement: Cross-Country Analysis Using Luxembourg Income Study Surveys. Soc. Sci. Q.
**2020**, 101, 712–731. [Google Scholar] [CrossRef] - Jorda, V.; Sarabia, J.M.; Jäntti, M. Estimation of income inequality from grouped data. arXiv
**2018**, arXiv:1808.09831. [Google Scholar] - Jenkins, S.P. Pareto models, top incomes and recent trends in UK income inequality. Economica
**2017**, 84, 261–289. [Google Scholar] [CrossRef] [Green Version] - Andreoli, F.; Zoli, C. Measuring Dissimilarity; Working Paper Series; Department of Economics University of Verona: Verona, Italy, 2014; Volume 23. [Google Scholar]
- Andreoli, F.; Zoli, C. From unidimensional to multidimensional inequality: A review. Metron
**2020**, 78, 5–42. [Google Scholar] [CrossRef] - Stiglitz, J.E.; Sen, A.; Fitoussi, J.P. Report by the Commission on the Measurement of Economic Performance and Social Progress; Citeseer: University Park, PA, USA, 2009. [Google Scholar]
- Hlasny, V.; AlAzzawi, S. Asset inequality in the MENA: The missing dimension? Q. Rev. Econ. Financ.
**2019**, 73, 44–55. [Google Scholar] [CrossRef] - Balarajan, Y.; Ramakrishnan, U.; Ozaltin, E.; Shankar, A.H.; Subramanian, S.V. Anaemia in low-income and middle-income countries. Lancet
**2011**, 378, 2123–2135. [Google Scholar] [CrossRef] - Scanlon, K.S.; Yip, R.; Schieve, L.A.; Cogswell, M.E. High and low hemoglobin levels during pregnancy: Differential risks for preterm birth and small for gestational age. Obstet. Gynecol.
**2000**, 96, 741–748. [Google Scholar] [CrossRef]

**Figure 1.**Bivariate Lorenz surfaces: evolution of multidimensional inequality in wealth and health. The curves are defined by Equation (14) with components defined in Equations (20) and (21).

Income | Health | Dependence | |||
---|---|---|---|---|---|

${\widehat{\mathit{a}}}_{\mathbf{1}}$ | ${\widehat{\mathit{b}}}_{\mathbf{1}}$ | ${\widehat{\mathit{a}}}_{\mathbf{2}}$ | ${\widehat{\mathit{b}}}_{\mathbf{2}}$ | $\mathit{w}$ | |

Angola | 1.2660 | 2.0952 | 17.8318 | 29.8269 | 3.6167 |

(0.0188) | (0.0438) | (1.2273) | (4.2834) | (0.7054) | |

Ethiopia | 2.8269 | 5.2741 | 0.2338 | 0.6067 | 2.3398 |

(0.0758) | (0.335) | (0.0147) | (0.027) | (0.114) | |

Haiti | 1.5794 | 3.8706 | 0.3814 | 1.8932 | 2.8026 |

(0.0284) | (0.0855) | (0.0231) | (0.0832) | (0.1965) | |

Nigeria | 0.4532 | 3.1033 | 0.1324 | 0.3306 | 1.1249 |

(0.012) | (0.0684) | (0.0068) | (0.0144) | (0.1341) | |

Uganda | 0.9473 | 2.2020 | 0.1994 | 0.6314 | 1.0116 |

(0.0185) | (0.0539) | (0.0211) | (0.0534) | (0.2055) |

Inequality | Equality | |||||
---|---|---|---|---|---|---|

Wealth | Health | Bidimensional | Total | Within | Between | |

Gini | Variables | Variables | ||||

Angola | 0.3516 | 0.1048 | 0.3943 | 0.6057 | 0.5766 | 0.0291 |

Ethiopia | 0.2570 | 0.6260 | 0.6921 | 0.3079 | 0.2608 | 0.0471 |

Haiti | 0.3467 | 0.6214 | 0.7273 | 0.2727 | 0.2294 | 0.0434 |

Nigeria | 0.6094 | 0.6678 | 0.8640 | 0.1360 | 0.1174 | 0.0186 |

Uganda | 0.4198 | 0.6680 | 0.7954 | 0.2046 | 0.1799 | 0.0247 |

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Sarabia, J.M.; Jorda, V.
Lorenz Surfaces Based on the Sarmanov–Lee Distribution with Applications to Multidimensional Inequality in Well-Being. *Mathematics* **2020**, *8*, 2095.
https://doi.org/10.3390/math8112095

**AMA Style**

Sarabia JM, Jorda V.
Lorenz Surfaces Based on the Sarmanov–Lee Distribution with Applications to Multidimensional Inequality in Well-Being. *Mathematics*. 2020; 8(11):2095.
https://doi.org/10.3390/math8112095

**Chicago/Turabian Style**

Sarabia, José María, and Vanesa Jorda.
2020. "Lorenz Surfaces Based on the Sarmanov–Lee Distribution with Applications to Multidimensional Inequality in Well-Being" *Mathematics* 8, no. 11: 2095.
https://doi.org/10.3390/math8112095