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A Class of Generalized Beta Distributions, Pareto Power Series and Weibull Power Series

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A Class of Generalized Beta Distributions, Pareto Power Series and Weibull Power Series A CLASS OF GENERALIZED BETA DISTRIBUTIONS, PARETO POWER SERIES AND WEIBULL POWER SERIES ALICE LEMOS DE MORAIS Primary advisor: Prof. Audrey Helen M. A. Cysneiros Secondary advisor: Prof. Gauss Moutinho Cordeiro Concentration area: Probability Disserta¸c˜ao submetida como requerimento parcial para obten¸c˜aodo grau de Mestre em Estat´ısticapela Universidade Federal de Pernambuco. Recife, fevereiro de 2009 Morais, Alice Lemos de A class of generalized beta distributions, Pareto power series and Weibull power series / Alice Lemos de Morais - Recife : O Autor, 2009. 103 folhas : il., fig., tab. Dissertação (mestrado) – Universidade Federal de Pernambuco. CCEN. Estatística, 2009. Inclui bibliografia e apêndice. 1. Probabilidade. I. Título. 519.2 CDD (22.ed.) MEI2009-034 Agradecimentos A` minha m˜ae,Marcia, e ao meu pai, Marcos, por serem meus melhores amigos. Agrade¸co pelo apoio `aminha vinda para Recife e pelo apoio financeiro. Agrade¸coaos meus irm˜aos, Daniel, Gabriel e Isadora, bem como a meus primos, Danilo e Vanessa, por divertirem minhas f´erias. A` minha tia Gilma pelo carinho e por pedir `asua amiga, Mirlana, que me acolhesse nos meus primeiros dias no Recife. A` Mirlana por amenizar minha mudan¸cade ambiente fazendo-me sentir em casa durante meus primeiros dias nesta cidade. Agrade¸comuito a ela e a toda sua fam´ılia. A` toda minha fam´ılia e amigos pela despedida de Belo Horizonte e `aS˜aozinha por preparar meu prato predileto, frango ao molho pardo com angu. A` vov´oLuzia e ao vovˆoTunico por toda amizade, carinho e preocupa¸c˜ao, por todo apoio nos meus dias mais dif´ıceis em Recife e pelo apoio financeiro nos momentos que mais precisei. A` Michelle, ao F´abioe `aMaristela por fazerem da ida ao Jardim de Minas, carinhosa- mente apelidado de JD, um dos meus compromissos mais esperados toda vez que retornava a Belo Horizonte. Agrade¸co`aTalita por toda amizade e por toda felicidade que ela me traz. Agrade¸co`aMichelle por me visitar no Recife. Ao professor Andrei Toom por ter me ensinado mais sobre lecionar que ele possa imaginar. Ao professor Klaus por ser um dos melhores professores que j´ative - pra n˜ao dizer o melhor. Ao Wagner, simplesmente por ser quem ele ´e,amigo, verdadeiro, atencioso, honesto, honrado. Agrade¸coa ele por ser uma das raz˜oes que mais fez valer a pena minha vinda para Recife. Agrade¸copor cuidar de mim, pelas partidas de domin´o,pelos dias e noites de estudo, por se empaturrar de polenguinho comigo, por me acompanhar na decis˜aos´ubita de ir a Natal s´opra ver o Atl´eticoMineiro jogar. Agrade¸coaos seus pais e `asua irm˜apelo carinho e por me fazer sentir t˜ao querida. A` Olga, `aRafaella e `aCl´audia, companheiras de moradia, pela amizade. Agrade¸co`a Juliana, Olga, Wagner e L´ıdiapela companhia em estudos e madrugadas fazendo trabalho. Agrade¸coaos demais colegas da turma de mestrado, Izabel, Wilton, Andr´ea,C´ıceroe Ma- noel. Agrade¸coaos colegas da estat´ıstica, Raphael, Marcelo, Lilian, Valmir, Hem´ılio,Abra˜ao, F´abioVer´ıssimo, Daniel, entre muitos outros que fizeram meus dias mais alegres. Agrade¸co ao Wagner, Alessandro, Rodrigo, Alexandre e `aAndr´ea, por muitas ajudas, sinucas e gar- galhadas. Ao Ives pelas visitas, pela divers˜aoe pelas ajudas. A` Val´eriapor toda ajuda. Aos amigos que conheci na computa¸c˜ao,especialmente Kalil, Rafael e seu amigo Luiz, por me divertirem tanto. Agrade¸co aos colegas da f´ısica,tamb´em,por me fazerem rir muito. Dentre eles agrade¸coparticularmente ao Vladimir, pela companhia, carinho e amizade ´ımpares. A` Anete pela amizade e pela deliciosa viagem para Pipa, assim como agrade¸coao Allan e `afam´ıliado Raphael por me receberem em suas casas em Natal. Agrade¸coaos pais e `a irm˜ado Raphael pelos passeios e pela batata da Barbie. Aos meus amigos de Belo Horizonte pela amizade, apoio e por me fazerem t˜ao bem. A` CAPES pela bolsa de estudos e ao Departamento de Estat´ısticapor me oferecerem este curso de mestrado. A` minha orientadora, Audrey Cysneiros, pela orienta¸c˜aoe por ajudar a me organizar. Agrade¸coao Gauss Cordeiro, que gentilmente aceitou me co-orientar. Agrade¸copelo que me ensinou e pelo que eu pude aprender somente ao observ´a-lo. Agrade¸co por ter encontrado nele um grande e raro exemplo de pessoa e profissional. Agrade¸copor ter me levado ao m´edicoquando n˜ao estive bem e por toda preocupa¸c˜ao, seja ela acadˆemicaou n˜ao. Ao Leandro Rˆegoe ao Borko Stosic, por terem aceitado o convite de participar da minha banca. Enfim, agrade¸coa todos que de alguma forma, direta ou indireta, fizeram dos meus dias neste mestrado e nesta cidade mais tranquilos e alegres. Resumo Nesta disserta¸c˜aotrabalhamos com trˆesclasses de distribui¸c˜oesde probabilidade, sendo uma j´aconhecida na literatura, a Classe de Distribui¸c˜oes Generalizadas Beta (Beta-G) e duas outras novas classes introduzidas nesta tese, baseadas na composi¸c˜ao das distribui¸c˜oes Pareto e Weibull com a classe de distribui¸c˜oesdiscretas power series. Fazemos uma revis˜ao geral da classe Beta-G e introduzimos um caso especial, a distribui¸c˜aobeta log´ıstica gen- eralizada do tipo IV (BGL(IV)). Introduzimos distribui¸c˜oesrelacionadas `aBGL(IV) que tamb´empertencem `aclasse Beta-G, como a beta-beta prime e a beta-F. Introduzimos a classe Pareto power series (PPS), que ´euma mistura de distribui¸c˜oes Pareto com pesos definidos pela distribui¸c˜aopower series, e apresentamos algumas de suas propriedades. In- troduzimos a classe Weibull power series (WPS), cujo processo de constru¸c˜ao´esimilar ao da classe PPS. Apresentamos algumas de suas propriedades e aplica¸c˜ao a um banco de dados reais. Distribui¸c˜oes nesta classe tˆem aplica¸c˜aointeressante a dados de tempo de vida devido `avariedade de formas da fun¸c˜aode risco. Para as classes PPS e WPS, fizemos uma sim- ula¸c˜ao para avaliar m´etodos de sele¸c˜ao de modelo. A distribui¸c˜ao pareto ´eum caso especial limite da distribui¸c˜ao PPS, assim como a distribui¸c˜ao Weibull ´eum caso especial limite da distribui¸c˜aoWPS. Palavras-chave: Distribui¸c˜oes generalizadas; Distribui¸c˜ao Beta; Distribui¸c˜aoWeibull; Distribui¸c˜ao de Pareto; Power series; Algoritmo EM. Abstract In this thesis we work with three classes of probability distributions, one already known in the literature, the Class of Generalized Beta Distributions (Beta-G) and two new classes, based on the composition of the Pareto and Weibull distributions and the class of discrete distributions power series. We make a general review of this class and introduced a special case, the beta generalized logistics of type IV (BGL(IV)) distribution. We introduce some distributions related to the BGL(IV) that are in the Beta-G class, such as beta-beta prime and beta-F distributions. We introduce the Pareto power series (PPS) class of distribu- tions, which is a mixture of the Pareto distribution with weights defined by power series, and present some of their properties. We introduce the Weibull power series (WPS) class of distributions, whose construction process is similar to the PPS class. We present some of its properties and application to a real data set. Distributions in this class have interesting ap- plication to lifetime data due to the variety of the shapes of the hazard function. The Pareto distribution is a limiting special case of the PPS distribution and the Weibull distribution is a special limit case of the WPS distribution. For the PPS and WPS classes we made a simulation to evaluate methods for model selection. Palavras-chave: Generalized distributions; Beta distribution; Weibull distribution; Pareto distribution; Power series; EM algorithm. Contents 1 Introduction 13 1.1 Apresenta¸c˜ao Inicial . 13 1.2 Initial presentation . 15 2 General Definition of the Class of Generalized Beta Distributions 16 2.1 Resumo . 16 2.2 Introduction . 16 2.3 Definition . 17 2.4 Density function . 17 2.5 Survival and hazard functions . 20 2.6 Moments . 21 2.7 Random number generation . 21 3 Some Special Distributions of the Class of Generalized Beta Distributions 23 3.1 Resumo . 23 3.2 Beta normal . 23 3.2.1 The distribution . 23 3.2.2 Moments . 25 3.2.3 Random number generation . 25 3.2.4 Estimation . 27 3.2.5 Application . 27 3.3 Beta-exponential . 28 3.3.1 The distribution . 28 3.3.2 Hazard function . 28 3.3.3 Moments . 30 3.3.4 Estimation . 30 3.3.5 Random number generation . 31 3.3.6 Application . 31 3.4 Beta Weibull . 33 3.4.1 The distribution . 33 3.4.2 Hazard function . 34 3.4.3 Moments . 35 3.4.4 Estimation . 35 7 3.4.5 Random number generation . 36 3.4.6 Application . 37 3.5 Beta-hyperbolic secant . 37 3.5.1 The distribution . 37 3.5.2 Estimation . 38 3.5.3 Random number generation . 39 3.5.4 Application . 40 3.6 Beta gamma . 40 3.6.1 The distribution . 41 3.6.2 Hazard function . 42 3.6.3 Moments . 43 3.6.4 Estimation . 43 3.6.5 Random number generation . 44 3.6.6 Application . 44 3.7 Beta Gumbel . 45 3.7.1 The distribution . 45 3.7.2 Estimation . 46 3.7.3 Random number generation . 47 3.7.4 Application . 47 3.8 Beta Fr´echet . 48 3.8.1 The distribution . 49 3.8.2 Hazard function . 49 3.8.3 Estimation . 50 3.8.4 Random number generation .
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