Extended circular distributions: mathematical properties, inference and regression model

• Autor(a) principal: PAULA, Fernanda Vital de
• 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
• topic: Estatística; Modelo de regressão

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.