Primjena Lorenzove krivulje na distribuciju prihoda

master's thesis

• Author: Marina Marjanović-Zulim
• Mentor: Siniša Slijepčević (mentor)
• Committee member: Siniša Slijepčević (predsjednik povjerenstva)
• Committee member: Miljenko Marušić (član povjerenstva)
• Committee member: Dragutin Svrtan (član povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2014-09-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu smo se usredotočili na opisivanje distribucija prihoda i Lorenzovih krivulja. Prvo poglavlje započinje modelom za distribuciju prihoda koju su predložili Singh i Maddala (1976), koji je generalizacija Paretovog i Weibullovog modela. Model je troparametarska distribucija prihoda koja se temelji na stopi neuspjeha. Model ima široku primjenu u empirijskim studijima kao model za distribuciju prihoda koji jako dobro opisuje podatke različitih zemalja. U nastavku, iznosimo novi model za distribucije prihoda kojeg je objavio Camilo Dagum. Taj model je posljedica skupa vrlo važnih pretpostavki. Ovaj model kasnije postane poznat kao Dagumova distribucija koja se široko primjenjuje u empirijskim studijima za opisivanje distribucija prihoda. Drugo poglavlje je izlaganje modela za opisivanje distribucija prihoda. Počinje sa generaliziranom beta distribucijom koja ima četiri parametra. Pokazano je da beta distribucija uključuje beta distribucije prve i druge vrste, Singh-Maddala, log-normalnu, gama, Weibullovu i eksponencijalnu distribuciju kao specijalne ili granične slučajeve. Zatim pruža pregled Paretovog modela, te hijerarhiju generaliziranih Paretovih modela. Dana su svojstva tih modela, te njihovi međuodnosi. Nakon toga, poglavlje predstavlja Dagumove distribucije, te njihove međuodnose sa drugim statističkim distribucijama. Poglavlje završava međuodnosima svih prikazanih modela. Poglavlje tri proučava distribuciju BDP-a po stanovniku za populaciju od 120 zemalja kroz period od 1960: do 1989: godine. Uočeno je kako su distribucije bimodalne. Proučavani podaci su na kraju opisani mješovitom distribucijom koja je nastala kombinacijom Weibullove i odrezane normalne distribucije. U četvrtom poglavlju govori se o Lorenzovoj krivulji. Poglavlje uključuje osnovna svojstva krivulje, te prikaz Lorenzovih krivulja za distribucije opisane u prethodnom poglavlju. Predstavljen je i opći model za izvođenje hijerarhijske familije Lorenzovih krivulja. Također, donosi i najpoznatiju mjeru nejednakosti za opisane distribucije, Gini koeficijent. Poglavlje završava Lorenzovim uređajem.

Abstract (english)

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.

• Keywords: modeli za distribuciju prihoda; Dagumova distribucija; Paretov model; Lorenzova krivulja
• Keywords (english): models for income distribution; Dagum distributions; Pareto models; Lorenz curve
• Language: croatian
• URN:NBN: urn:nbn:hr:217:896914
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Access restricted to students and staff of home institution
• Created on: 2019-01-22 10:50:19