# 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 |

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**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