## Modeling Income Distributions and Lorenz CurvesDuangkamon Chotikapanich Jean-Jacques Rousseau wrote in the Preface to his famous Discourse on Inequality that “I consider the subject of the following discourse as one of the most interesting questions philosophy can propose, and unhappily for us, one of the most thorny that philosophers can have to solve. For how shall we know the source of inequality between men, if we do not begin by knowing mankind?” (Rousseau, 1754). This citation of Rousseau appears in an article in Spanish where Dagum (2001), in the memory of whom this book is published, also cites Socrates who said that the only useful knowledge is that which makes us better and Seneca who wrote that knowing what a straight line is, is not important if we do not know what rectitude is. These references are indeed a good illustration of Dagum’s vast knowledge, which was clearly not limited to the ?eld of Economics. For Camilo the ?rst part of Rousseau’s citation certainly justi?ed his interest in the ?eld of inequality which was at the centre of his scienti?c preoccupations. It should however be stressed that for Camilo the second part of the citation represented a “solid argument in favor of giving macroeconomic foundations to microeconomic behavior” (Dagum, 2001). More precisely, “individualism and methodological holism complete each other in contributing to the explanation of individual and social behavior” (Dagum, 2001). |

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### Contents

3 | |

A Function for Size Distribution of Incomes | 27 |

Some Generalized Functions for the Size Distribution of Income | 37 |

Efﬁcient Estimation of the Lorenz Curve and Associated Inequality Measures from Grouped Observations | 57 |

Distribution and Mobility of Wealth of Nations | 71 |

Survey papers on Lorenz functions and the generalizations and extensions of income distributions | 96 |

A Guide to the Dagum Distributions | 97 |

Pareto and Generalized Pareto Distributions | 119 |

Maximum Entropy Estimation of Income Distributions from Bonferroni Indices | 193 |

New Four and FiveParameter Models for Income Distributions | 211 |

Fuzzy Monetary Poverty Measures under a Dagum Income Distributive Hypothesis | 225 |

Modelling Lorenz Curves Robust and Semiparametric Issues | 241 |

Modelling Inequality with a Single Parameter | 255 |

Lorenz Curves and Generalised Entropy Inequality Measures | 271 |

Estimating Income Distributions Using a Mixture of Gamma Densities | 285 |

Inequality in Multidimensional Indicators of WellBeing Methodology and Application to the Human Development Index | 303 |

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analysis approach Bandourian beta distribution bution Chotikapanich classical Pareto components computed corresponding cumulative distribution function Dagum distribution Dagum model Dagum type decile deﬁned deﬁnition denotes density function derived distri distribution function Distribution of Income dPlN Econometrica Econometrics Economic efﬁcient empirical entropy equation exponential ﬁrst ﬁt ﬁtted ﬁtting ﬂexibility functional form gamma gamma distribution GDP per capita Gini coefﬁcient Gini index given income data income distribution income groups income inequality indices inequality measures Journal Kakwani Kleiber Lemmi likelihood function lognormal distribution Lorenz curve Lorenz ordering Maddala McDonald mean method mixture Model of Income observations obtained order statistics paper Pareto distribution Pareto model Podder poor group posterior probability proposed random variables ratio real GDP rich group sample Sarabia Singh-Maddala Speciﬁcation Statistics stochastic Stochastic Dominance subperiod Table Theil tion values variance Victoria-Feser Weibull Weibull distribution