# Primjena Lorenzove krivulje na distribucije prihoda

Marjanović-Zulim, Marina (2014) Primjena Lorenzove krivulje na distribucije prihoda. Diploma thesis, Faculty of Science > Department of Mathematics.

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## Abstract

The focus of this paper is on modeling income distributions and Lorenz curves. The volume is organized in four parts. Chapter one begins with model for income distribution proposed by Singh and Maddala (1976), that is generalization of Pareto and Weibull models. The model is a three parameter income distribution and is based on the concept of failure rate. This model is also used widely in empirical studies as an income distribution model that very well describes the data from various countries. Further on, we present the 1977 paper by Camilo Dagum on a new model for the size distribution of incomes that answers to a set of important assumptions. Dagum established empirical foundations in the form of properties for a probability function to describe the size distribution of income. This model later became known as the Dagum distribution and is now widely used in empirical studies as one of the models that well represents income distributions. Chapter two is a presentation of models for the size distribution of incomes. It starts with generalized beta distribution, a four parameter distribution. It was shown that beta includes the beta of the 1st kind, the beta of the 2nd kind, the Singh-Maddala, the lognormal, gamma, Weibull and exponential distributions as special or limiting cases. Then it provides a survey on the classical Pareto model and a hierarchy of generalized Pareto models. The properties of these models are introduced where the related distributions and inferential issues are discussed. After that, this part introduces the Dagum distributions and their interrelations with other statistical distributions. It provides the basic statistical properties and inferential aspects of the Dagum distributions. Chapter ends with interrelations of presented distributions. Chapter three is a study on the distributions of real GDP per capita for a combined 120 countries over the period from 1960 to 1989. These distributions appear to be bimodal. In this chapter a mixture of Weibull and truncated normal densities are used to model the bimodal distributions. The fourth chapter is on the Lorenz curve. This chapter includes the basic properties of Lorenz curve and its representation of models presented in the previous chapter. A general method for obtaining a hierarchical family of Lorenz curves is introduced. It derives inequality measure, the Gini index of described distributions. The chapter ends with Lorenz order.

Item Type: Thesis (Diploma thesis) Slijepčević, Siniša 2014 51 NATURAL SCIENCES > Mathematics Faculty of Science > Department of Mathematics Iva Prah 05 Jun 2015 11:05 05 Jun 2015 11:05 http://digre.pmf.unizg.hr/id/eprint/4003

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