Inferência e Modelação Estatística Probabilidades, Estatística e Modelos Lineares Generalizados
Subtitle: Probabilidades, Estatística e Modelos Lineares Generalizados
Edition: First Edition
Publisher: Nova.FCT Editorial
Published: January 2018
Place of publication: Portugal
Language: por
Format: Paperback
Number of pages: 134
ISBN-13: 9789899952898
Copyright date: Second printing: December de 2020
First sentence: A teoria das probabilidades evoluiu recentemente – se considerarmos a escala de tempo
definida pela evolução da geometria – tomando-se como ponto de partida as contribuições
iniciais de Girolamo Cardamo (1501–1576), Pierre de Fermat (1601–1665) e
Blaise Pascal (1623 – 1662).
Description: This text aims to present an introduction to some important topics in Mathematical Statistics and their applications. The text was developed from notes of lessons from the curricular unit Inference and Statistical Modeling, from the second year of the Masters in Mathematics and Applications at FCT Nova da UNL.
The first chapter reviews some of the main results of the modern theory of Probability – according to Andrei Kolmogorov – that will be necessary for the study of Statistical Inference. In the second chapter, the moment generating functions are introduced with two objectives: the first is to apply the Lebesgue integral and its essential operative properties to the calculation of expected values; the second is the study of the strong law of large numbers – via the Cramer-Chernov large deviation theorem – and of some limit theorems – for example the central limit theorem – for random variables that admit a non-trivial moment generating function. In the third chapter, using the formalism introduced previously, we explore the main concepts of Statistical Inference. In the part of the hypothesis tests, the approximation by the likelihood ratios is privileged. In the fourth chapter, generalised linear models are studied in a practical perspective but supported by the statistical formalism studied.
Table of contents: title: 1 Formalismo das Probabilidades 1; title: 1.1 Introdução . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1; title: 1.2 O modelo das probabilidades segundo Kolmogorov . . . . . . . . . . . 2; title: 1.3 Leis dos grandes números . . . . . . . . . . . . . . . . . . . . . . . . . 9; title: 1.4 Um teorema limite central . . . . . . . . . . . . . . . . . . . . . . . . . 12; title: 1.5 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14; title: 1.6 Resoluções . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18; title: 2 FGM e Teoremas Limite 23; title: 2.1 Introdução . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23; title: 2.2 Sobre o cálculo de valores esperados . . . . . . . . . . . . . . . . . . . 23; title: 2.3 Os momentos das variáveis aleatórias . . . . . . . . . . . . . . . . . . . 24; title: 2.4 A função geradora de momentos . . . . . . . . . . . . . . . . . . . . . 26; title: 2.5 Propriedades da função geradora de momentos . . . . . . . . . . . . . 30; title: 2.6 Teoremas limite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33; title: 2.7 Grandes desvios e lei forte dos grandes números . . . . . . . . . . . . . 43; title: 2.8 Apêndice: sobre as funções convexas . . . . . . . . . . . . . . . . . . . 46; title: 2.9 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53; title: 2.10 Resoluções . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55; title: 3 Formalismo da Estatística 57; title: 3.1 Introdução . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57; title: 3.2 Estatísticas e estimadores . . . . . . . . . . . . . . . . . . . . . . . . . 58; title: 3.3 Os Estimadores da Máxima Verosimilhança . . . . . . . . . . . . . . . 63; title: 3.4 Informação de Fisher . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68; title: 3.5 Consistência e Normalidade Assimptótica dos EMV . . . . . . . . . . 71; title: 3.6 Estatísticas Suficientes . . . . . . . . . . . . . . . . . . . . . . . . . . . 74; title: 3.7 Testes de Hipóteses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75; title: 3.8 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84; title: 3.9 Resoluções . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88; title: 4 Modelos Lineares Generalizados 95; title: 4.1 Introdução . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95; title: 4.2 Distribuições de tipo exponencial . . . . . . . . . . . . . . . . . . . . . 96; title: 4.3 A função de ligação canónica . . . . . . . . . . . . . . . . . . . . . . . 100; title: 4.4 A estimação nos modelos lineares generalizados . . . . . . . . . . . . . 101; title: 4.5 Métodos para aferir a qualidade do ajustamento . . . . . . . . . . . . 102; title: 4.6 O modelo Logit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107; title: xi; title: xii Índice; title: 4.7 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113; title: 4.8 Resoluções . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114; title: 4.9 Comentários sobre as referências . . . . . . . . . . . . . . . . . . . . . 116; title: Bibliografia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117