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An Introduction to Mathematical Statistics Fetsje Bijma

An Introduction to Mathematical Statistics By Fetsje Bijma

An Introduction to Mathematical Statistics by Fetsje Bijma

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Summary

This book gives an introduction into mathematical statistics.

An Introduction to Mathematical Statistics Summary

An Introduction to Mathematical Statistics by Fetsje Bijma

Statistics is the science that focuses on drawing conclusions from data, by modeling and analyzing the data using probabilistic models. In An Introduction to Mathematical Statistics the authors describe key concepts from statistics and give a mathematical basis for important statistical methods. Much attention is paid to the sound application of those methods to data.

The three main topics in statistics are estimators, tests, and confidence regions. The authors illustrate these in many examples, with a separate chapter on regression models, including linear regression and analysis of variance. They also discuss the optimality of estimators and tests, as well as the selection of the best-fitting model.

Each chapter ends with a case study in which the described statistical methods are applied. This book assumes a basic knowledge of probability theory, calculus, and linear algebra.

Several annexes are available for Mathematical Statistics on this page.

About Fetsje Bijma

Fetsje Bijma worked as an assistant professor of mathematics at the Vrije Universiteit Amsterdam for 10 years. She studied mathematics at the University of Groningen and received a Ph.D. at the Faculty of Medicine of the Vrije Universiteit Amsterdam. She now works in industry. Marianne Jonker is a biostatistician at the Radboud University Medical Center in Nijmegen. She studied mathematics and received her Ph.D. at the Vrije Universiteit Amsterdam, where she worked as an assistant professor in mathematics for some time. She has done work as a biostatistician for the Leiden University Medical Center in Leiden and the VU University Medical Center in Amsterdam. Aad van der Vaart is a professor of stochastics at Leiden University and member of the Royal Netherlands Academy of Arts and Sciences. He studied mathematics and psychology at Leiden University and received a Ph.D. in mathematics at the same university. He held positions at universities in the United States and in Paris (France) and at the Vrije Universiteit Amsterdam. In 2015 he won the highest scientific award in the Netherlands, the NWO Spinoza Prize, for his groundbreaking research in statistics.

Table of Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1. WhatIsStatistics? . . . . . . . . . . . . . . . . . . . . . 1 1.2. StatisticalModels . . . . . . . . . . . . . . . . . . . . . 2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 12 Application: Cox Regression . . . . . . . . . . . . . . . . . 15 2. DescriptiveStatistics . . . . . . . . . . . . . . . . . . . . . . 21 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2. UnivariateSamples . . . . . . . . . . . . . . . . . . . . . 21 2.3. Correlation . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . 38 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 39 Application: Benford's Law . . . . . . . . . . . . . . . . . 41 3. Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2. MeanSquareError . . . . . . . . . . . . . . . . . . . . . 46 3.3. Maximum Likelihood Estimators . . . . . . . . . . . . . . . 54 3.4. MethodofMomentsEstimators . . . . . . . . . . . . . . . . 72 3.5. BayesEstimators . . . . . . . . . . . . . . . . . . . . . . 75 3.6. M-Estimators . . . . . . . . . . . . . . . . . . . . . . . 88 3.7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . 93 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 94 Application: Twin Studies . . . . . . . . . . . . . . . . . 100 4. Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . 105 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 105 4.2. Null Hypothesis and Alternative Hypothesis . . . . . . . . . . 105 4.3. SampleSizeandCriticalRegion . . . . . . . . . . . . . . 107 4.4. Testing with p-Values . . . . . . . . . . . . . . . . . . . 121 4.5. StatisticalSignificance . . . . . . . . . . . . . . . . . . 126 4.6. SomeStandardTests . . . . . . . . . . . . . . . . . . . 127 4.7. Likelihood Ratio Tests . . . . . . . . . . . . . . . . . . 143 4.8. ScoreandWaldTests . . . . . . . . . . . . . . . . . . . 150 4.9. Multiple Testing . . . . . . . . . . . . . . . . . . . . . 153 4.10. Summary . . . . . . . . . . . . . . . . . . . . . . . . 159 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 160 Application: Shares According to Black-Scholes . . . . . . . . 169 5. ConfidenceRegions . . . . . . . . . . . . . . . . . . . . . 174 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 174 5.2. Interpretation of a Confidence Region . . . . . . . . . . . . 174 5.3. PivotsandNear-Pivots . . . . . . . . . . . . . . . . . . 177 5.4. Maximum Likelihood Estimators as Near-Pivots . . . . . . . . 181 5.5. ConfidenceRegionsandTests . . . . . . . . . . . . . . . 195 5.6. Likelihood Ratio Regions . . . . . . . . . . . . . . . . . 198 5.7. BayesianConfidenceRegions . . . . . . . . . . . . . . . . 201 5.8. Summary . . . . . . . . . . . . . . . . . . . . . . . . 205 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 206 Application: The Salk Vaccine . . . . . . . . . . . . . . . 209 6. Optimality Theory . . . . . . . . . . . . . . . . . . . . . . 212 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 212 6.2. SufficientStatistics . . . . . . . . . . . . . . . . . . . . 212 6.3. EstimationTheory . . . . . . . . . . . . . . . . . . . . 219 6.4. TestingTheory . . . . . . . . . . . . . . . . . . . . . 231 6.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . 245 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 246 Application: High Water in Limburg . . . . . . . . . . . . . 250 7. RegressionModels . . . . . . . . . . . . . . . . . . . . . . 259 7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 259 7.2. LinearRegression . . . . . . . . . . . . . . . . . . . . 261 7.3. AnalysisofVariance . . . . . . . . . . . . . . . . . . . 275 7.4. Nonlinear and Nonparametric Regression . . . . . . . . . . . 283 7.5. Classification . . . . . . . . . . . . . . . . . . .

Additional information

NGR9789462985100
9789462985100
9462985103
An Introduction to Mathematical Statistics by Fetsje Bijma
Fetsje Bijma
New
Paperback
Amsterdam University Press
20170515
384
N/A
Book picture is for illustrative purposes only, actual binding, cover or edition may vary.
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