Extended circular distributions: mathematical properties, inference and regression model
Autor(a) principal: | |
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Data de Publicação: | 2018 |
Tipo de documento: | Tese |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UFPE |
Texto Completo: | https://repositorio.ufpe.br/handle/123456789/29752 |
Resumo: | Circular Statistics is an important branch of the Statistics which has been necessary in various scientific fields such as Biology, Medicine, Geology, Meteorology and others. During the performed researches some gaps were observed in this study branch. Thus, the main objective of this thesis is to collaborate with the enrichment of the literature in Circular Statistics, seeking to fill these gaps. First, the difficulty in obtaining models with asymmetry, different modality scenarios and treatable trigonometric moments was noticed. In this way, a new circular distribution is proposed in the Chapter 2, denominated Exponentialized Cardioid (EC). Some of its mathematical properties are presented, such as trigonometric moments, kurtosis, and asymmetry. In addition, two estimation methods for EC model parameters were studied. Subsequently, the lack of hypothesis tests for parameters of circular distributions, in the context of models distinction, was evidenced in the literature. Apart from, few studies on bootstrap were found in Circular Statistics. Thus, in Chapter 3, we devote attention to make hypothesis inference on EC parameters. In particular, adopting as comparison critera estimated type I error size and test power, we study the performance of tests based on likelihood ratio, Wald, score and gradient statistics and their bootstrap versions putting emphasis to distinguish the EC distribution regard to Cardioid and uniform models, special cases of the former. From the theoretical point of view, an important collaboration was the derivation of the EC Fisher information matrix. The last gap refers to the few models of circular-circular regression in the literature. In Chapter 4, a new circular-circular regression model having distributed EC angular errors is proposed. Its regression curve is expressed in terms of the Möbius Transformation. Futher, a complex version of the EC distribution is also presented, named CEC distribution, and a likelihood-based estimation procedure for parameters of the new model is furnished. The fifth Chapter has the same purpose as Chapter 2. Four new circular distributions, that extend the Cardioid distribution (C) are proposed, called beta Cardioid (βC), Kumaraswamy Cardioid (KwC), gamma Cardioid (ΓC) and Marshall-Olkin Cardioid (MOC). These distributions are rewritten as a family, which is a result of weighting the C probability density function (pdf). General mathematical expressions for their trigonometric moments and the idea for estimating the parameters of the proposed models by the maximum likelihood method are presented. These four chapters present examples in the area of Meteorology or Biology that point out the success of the new proposed models in the Chapters 2, 4 and 5 and the good performance of the Wald and gradient tests, in the Chapter 3. |
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PAULA, Fernanda Vital dehttp://lattes.cnpq.br/9279556456110004http://lattes.cnpq.br/7674916684282039AMARAL, Getúlio José Amorim doNASCIMENTO, Abraão David Costa do2019-03-18T21:55:17Z2019-03-18T21:55:17Z2018-03-14https://repositorio.ufpe.br/handle/123456789/29752Circular Statistics is an important branch of the Statistics which has been necessary in various scientific fields such as Biology, Medicine, Geology, Meteorology and others. During the performed researches some gaps were observed in this study branch. Thus, the main objective of this thesis is to collaborate with the enrichment of the literature in Circular Statistics, seeking to fill these gaps. First, the difficulty in obtaining models with asymmetry, different modality scenarios and treatable trigonometric moments was noticed. In this way, a new circular distribution is proposed in the Chapter 2, denominated Exponentialized Cardioid (EC). Some of its mathematical properties are presented, such as trigonometric moments, kurtosis, and asymmetry. In addition, two estimation methods for EC model parameters were studied. Subsequently, the lack of hypothesis tests for parameters of circular distributions, in the context of models distinction, was evidenced in the literature. Apart from, few studies on bootstrap were found in Circular Statistics. Thus, in Chapter 3, we devote attention to make hypothesis inference on EC parameters. In particular, adopting as comparison critera estimated type I error size and test power, we study the performance of tests based on likelihood ratio, Wald, score and gradient statistics and their bootstrap versions putting emphasis to distinguish the EC distribution regard to Cardioid and uniform models, special cases of the former. From the theoretical point of view, an important collaboration was the derivation of the EC Fisher information matrix. The last gap refers to the few models of circular-circular regression in the literature. In Chapter 4, a new circular-circular regression model having distributed EC angular errors is proposed. Its regression curve is expressed in terms of the Möbius Transformation. Futher, a complex version of the EC distribution is also presented, named CEC distribution, and a likelihood-based estimation procedure for parameters of the new model is furnished. The fifth Chapter has the same purpose as Chapter 2. Four new circular distributions, that extend the Cardioid distribution (C) are proposed, called beta Cardioid (βC), Kumaraswamy Cardioid (KwC), gamma Cardioid (ΓC) and Marshall-Olkin Cardioid (MOC). These distributions are rewritten as a family, which is a result of weighting the C probability density function (pdf). General mathematical expressions for their trigonometric moments and the idea for estimating the parameters of the proposed models by the maximum likelihood method are presented. These four chapters present examples in the area of Meteorology or Biology that point out the success of the new proposed models in the Chapters 2, 4 and 5 and the good performance of the Wald and gradient tests, in the Chapter 3.A Estatística Circular é um ramo importante da Estatística que tem sido necessário em diversos campos científicos como Biologia, Medicina, Geologia, Meteorologia, entre outros. Durante as pesquisas realizadas foram observadas algumas lacunas neste campo de estudo. Dessa forma, o objetivo principal dessa tese é colaborar com o enriquecimento da literatura em Estatística Circular, buscando preencher tais lacunas. Primeiramente, a dificuldade em obter modelos com assimetria, diferentes cenários de modalidade e momentos trigonométricos tratáveis foi notada. Dessa forma, no Capítulo 2 uma nova distribuição circular é proposta, denominada Cardioide Exponencializada (EC). Algumas de suas propriedades matemáticas são apresentadas, como momentos trigonométricos, curtose e assimetria. Além disso, dois métodos de estimação para os parâmetros do EC modelo foram estudados. Posteriormente, a inexistência de testes de hipóteses para parâmetros de distribuições circulares, no contexto de distinção de modelos, foi evidenciada na literatura. Ademais, poucos estudos sobre bootstrap foram encontrados em Estatística Circular. Assim, no Capítulo 3, foi dada atenção à inferência de hipóteses sobre os parâmetros da EC. Em particular, adotando como critérios de comparação o tamanho do erro tipo I e o poder do teste estimados, estudamos o desempenho dos testes baseados na estatísticas de razão de verossimilhança, Wald, escore e gradiente e suas versões bootstrap, com ênfase em distinguir a distribuição EC dos modelos Cardioide (C) e uniforme, casos especiais da primeira. Do ponto de vista teórico, uma importante colaboração foi a derivação da matriz de informação de Fisher da EC. A última lacuna se refere aos poucos modelos de regressão circular-circular existentes na literatura. No Capítulo 4, um novo modelo de regressão circular-circular, com erros angulares assumindo a distribuição EC, é proposto. A curva de regressão é expressa em termos da transformação Möbius. Além disso, uma versão complexa da distribuição EC também é apresentada, denominada distribuição CEC e um procedimento, com base na máxima verossimilhança, é fornecido para estimar os parâmetros do novo modelo. O quinto capítulo tem o mesmo objetivo do Capítulo 2. Quatro novas distribuições circulares flexíveis que estendem a distribuição Cardioide (C) são propostas, denominadas beta Cardioide (βC), Kumaraswamy Cardioide (KwC), gamma Cardioide (ΓC) e Marshall-Olkin Cardioide (MOC). Estas distribuições são reescritas como ponderações da função densidade de probabilidade da C. As expressões matemáticas para seus momentos trigonométricos e a idéia geral para estimar os parâmetros dos modelos propostos pelo método de máxima verossimilhança são apresentadas. Esses quatro capítulos apresentam exemplos na área de Meteorologia e Biologia que apontam o sucesso dos novos modelos propostos nos Capítulos 2, 4 e 5 e o bom desempenho dos testes Wald e gradiente, no Capítulo 3.engUniversidade Federal de PernambucoPrograma de Pos Graduacao em EstatisticaUFPEBrasilAttribution-NonCommercial-NoDerivs 3.0 Brazilhttp://creativecommons.org/licenses/by-nc-nd/3.0/br/info:eu-repo/semantics/openAccessEstatísticaModelo de regressãoExtended circular distributions: mathematical properties, inference and regression modelinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisdoutoradoreponame:Repositório Institucional da UFPEinstname:Universidade Federal de Pernambuco (UFPE)instacron:UFPETHUMBNAILTESE Fernanda Vital de Paula.pdf.jpgTESE Fernanda Vital de Paula.pdf.jpgGenerated Thumbnailimage/jpeg1227https://repositorio.ufpe.br/bitstream/123456789/29752/5/TESE%20Fernanda%20Vital%20de%20Paula.pdf.jpg618d066df42e9cd171b19b4a88194a7aMD55ORIGINALTESE Fernanda Vital de Paula.pdfTESE Fernanda Vital de Paula.pdfapplication/pdf3234926https://repositorio.ufpe.br/bitstream/123456789/29752/1/TESE%20Fernanda%20Vital%20de%20Paula.pdf5fb3ba78e50adbe2eab13c4feab8b459MD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; 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dc.title.pt_BR.fl_str_mv |
Extended circular distributions: mathematical properties, inference and regression model |
title |
Extended circular distributions: mathematical properties, inference and regression model |
spellingShingle |
Extended circular distributions: mathematical properties, inference and regression model PAULA, Fernanda Vital de Estatística Modelo de regressão |
title_short |
Extended circular distributions: mathematical properties, inference and regression model |
title_full |
Extended circular distributions: mathematical properties, inference and regression model |
title_fullStr |
Extended circular distributions: mathematical properties, inference and regression model |
title_full_unstemmed |
Extended circular distributions: mathematical properties, inference and regression model |
title_sort |
Extended circular distributions: mathematical properties, inference and regression model |
author |
PAULA, Fernanda Vital de |
author_facet |
PAULA, Fernanda Vital de |
author_role |
author |
dc.contributor.authorLattes.pt_BR.fl_str_mv |
http://lattes.cnpq.br/9279556456110004 |
dc.contributor.advisorLattes.pt_BR.fl_str_mv |
http://lattes.cnpq.br/7674916684282039 |
dc.contributor.author.fl_str_mv |
PAULA, Fernanda Vital de |
dc.contributor.advisor1.fl_str_mv |
AMARAL, Getúlio José Amorim do |
dc.contributor.advisor-co1.fl_str_mv |
NASCIMENTO, Abraão David Costa do |
contributor_str_mv |
AMARAL, Getúlio José Amorim do NASCIMENTO, Abraão David Costa do |
dc.subject.por.fl_str_mv |
Estatística Modelo de regressão |
topic |
Estatística Modelo de regressão |
description |
Circular Statistics is an important branch of the Statistics which has been necessary in various scientific fields such as Biology, Medicine, Geology, Meteorology and others. During the performed researches some gaps were observed in this study branch. Thus, the main objective of this thesis is to collaborate with the enrichment of the literature in Circular Statistics, seeking to fill these gaps. First, the difficulty in obtaining models with asymmetry, different modality scenarios and treatable trigonometric moments was noticed. In this way, a new circular distribution is proposed in the Chapter 2, denominated Exponentialized Cardioid (EC). Some of its mathematical properties are presented, such as trigonometric moments, kurtosis, and asymmetry. In addition, two estimation methods for EC model parameters were studied. Subsequently, the lack of hypothesis tests for parameters of circular distributions, in the context of models distinction, was evidenced in the literature. Apart from, few studies on bootstrap were found in Circular Statistics. Thus, in Chapter 3, we devote attention to make hypothesis inference on EC parameters. In particular, adopting as comparison critera estimated type I error size and test power, we study the performance of tests based on likelihood ratio, Wald, score and gradient statistics and their bootstrap versions putting emphasis to distinguish the EC distribution regard to Cardioid and uniform models, special cases of the former. From the theoretical point of view, an important collaboration was the derivation of the EC Fisher information matrix. The last gap refers to the few models of circular-circular regression in the literature. In Chapter 4, a new circular-circular regression model having distributed EC angular errors is proposed. Its regression curve is expressed in terms of the Möbius Transformation. Futher, a complex version of the EC distribution is also presented, named CEC distribution, and a likelihood-based estimation procedure for parameters of the new model is furnished. The fifth Chapter has the same purpose as Chapter 2. Four new circular distributions, that extend the Cardioid distribution (C) are proposed, called beta Cardioid (βC), Kumaraswamy Cardioid (KwC), gamma Cardioid (ΓC) and Marshall-Olkin Cardioid (MOC). These distributions are rewritten as a family, which is a result of weighting the C probability density function (pdf). General mathematical expressions for their trigonometric moments and the idea for estimating the parameters of the proposed models by the maximum likelihood method are presented. These four chapters present examples in the area of Meteorology or Biology that point out the success of the new proposed models in the Chapters 2, 4 and 5 and the good performance of the Wald and gradient tests, in the Chapter 3. |
publishDate |
2018 |
dc.date.issued.fl_str_mv |
2018-03-14 |
dc.date.accessioned.fl_str_mv |
2019-03-18T21:55:17Z |
dc.date.available.fl_str_mv |
2019-03-18T21:55:17Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
format |
doctoralThesis |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://repositorio.ufpe.br/handle/123456789/29752 |
url |
https://repositorio.ufpe.br/handle/123456789/29752 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.rights.driver.fl_str_mv |
Attribution-NonCommercial-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nc-nd/3.0/br/ info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
Attribution-NonCommercial-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nc-nd/3.0/br/ |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Universidade Federal de Pernambuco |
dc.publisher.program.fl_str_mv |
Programa de Pos Graduacao em Estatistica |
dc.publisher.initials.fl_str_mv |
UFPE |
dc.publisher.country.fl_str_mv |
Brasil |
publisher.none.fl_str_mv |
Universidade Federal de Pernambuco |
dc.source.none.fl_str_mv |
reponame:Repositório Institucional da UFPE instname:Universidade Federal de Pernambuco (UFPE) instacron:UFPE |
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UFPE |
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UFPE |
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Repositório Institucional da UFPE |
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