Richard J. Rossi

Mathematical Statistics (eBook, PDF)

An Introduction to Likelihood Based Inference

• Format: PDF

Produktbeschreibung

Presents a unified approach to parametric estimation, confidence intervals, hypothesis testing, and statistical modeling, which are uniquely based on the likelihood function This book addresses mathematical statistics for upper-undergraduates and first year graduate students, tying chapters on estimation, confidence intervals, hypothesis testing, and statistical models together to present a unifying focus on the likelihood function. It also emphasizes the important ideas in statistical modeling, such as sufficiency, exponential family distributions, and large sample properties. Mathematical Statistics: An Introduction to Likelihood Based Inference makes advanced topics accessible and understandable and covers many topics in more depth than typical mathematical statistics textbooks. It includes numerous examples, case studies, a large number of exercises ranging from drill and skill to extremely difficult problems, and many of the important theorems of mathematical statistics along with their proofs. In addition to the connected chapters mentioned above, Mathematical Statistics covers likelihood-based estimation, with emphasis on multidimensional parameter spaces and range dependent support. It also includes a chapter on confidence intervals, which contains examples of exact confidence intervals along with the standard large sample confidence intervals based on the MLE's and bootstrap confidence intervals. There's also a chapter on parametric statistical models featuring sections on non-iid observations, linear regression, logistic regression, Poisson regression, and linear models. * Prepares students with the tools needed to be successful in their future work in statistics data science * Includes practical case studies including real-life data collected from Yellowstone National Park, the Donner party, and the Titanic voyage * Emphasizes the important ideas to statistical modeling, such as sufficiency, exponential family distributions, and large sample properties * Includes sections on Bayesian estimation and credible intervals * Features examples, problems, and solutions Mathematical Statistics: An Introduction to Likelihood Based Inference is an ideal textbook for upper-undergraduate and graduate courses in probability, mathematical statistics, and/or statistical inference.

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Produktdetails

• Verlag: John Wiley & Sons
• Seitenzahl: 448
• Erscheinungstermin: 8. Juni 2018
• Englisch
• ISBN-13: 9781118770979
• Artikelnr.: 53057481

Autorenporträt

Richard J. Rossi, PhD, is Director of the Statistics Program and Co-Director of the Data Science Program at Montana Tech of The University of Montana, in Butte, MT. He acted as President of the Montana Chapter of the American Statistical Association in 2001 and as Associate Editor for Biometrics from 1997-2000. Dr. Rossi is a member of the American Mathematical Society, the Institute of Mathematical Statistics, and the American Statistical Association.

Inhaltsangabe

Preface xiii

Acknowledgments xvii

1 Probability 1

1.1 Sample Spaces, Events, and sigma-Algebras 1

Problems 7

1.2 Probability Axioms and Rules 9

Problems 14

1.3 Probability with Equally Likely Outcomes 16

Problems 18

1.4 Conditional Probability 19

Problems 25

1.5 Independence 28

Problems 31

1.6 Counting Methods 33

Problems 38

1.7 Case Study -The Birthday Problem 41

Problems 44

2 Random Variables and Random Vectors 45

2.1 Random Variables 45

2.1.1 Properties of Random Variables 46

Problems 50

2.2 Random Vectors 53

2.2.1 Properties of Random Vectors 53

Problems 60

2.3 Independent Random Variables 63

Problems 66

2.4 Transformations of Random Variables 68

2.4.1 Transformations of Discrete Random Variables 68

2.4.2 Transformations of Continuous Random Variables 69

2.4.3 Transformations of Continuous Bivariate Random Vectors 73

Problems 75

2.5 Expected Values for Random Variables 77

2.5.1 Expected Values and Moments of Random Variables 77

2.5.2 The Variance of a Random Variable 81

2.5.3 Moment Generating Functions 86

Problems 89

2.6 Expected Values for Random Vectors 94

2.6.1 Properties of Expectation with Random Vectors 96

2.6.2 Covariance and Correlation 99

2.6.3 Conditional Expectation and Variance 106

Problems 110

2.7 Sums of Random Variables 114

Problems 120

2.8 Case Study - HowMany Times Was the Coin Tossed? 123

2.8.1 The Probability Model 124

Problems 126

3 Probability Models 129

3.1 Discrete Probability Models 129

3.1.1 The Binomial Model 129

3.1.1.1 Binomial Setting 130

3.1.2 The HypergeometricModel 132

3.1.2.1 Hypergeometric Setting 132

3.1.3 The Poisson Model 134

3.1.4 The Negative BinomialModel 135

3.1.4.1 Negative Binomial Setting 135

3.1.5 The MultinomialModel 138

3.1.5.1 Multinomial Setting 139

Problems 140

3.2 Continuous Probability Models 147

3.2.1 The Uniform Model 147

3.2.2 The Gamma Model 149

3.2.3 The Normal Model 152

3.2.4 The Log-normal Model 155

3.2.5 The Beta Model 156

Problems 158

3.3 Important Distributional Relationships 163

3.3.1 Sums of Random Variables 163

3.3.2 The T and F Distributions 166

Problems 170

3.4 Case Study -The Central LimitTheorem 172

3.4.1 Convergence in Distribution 172

3.4.2 The Central LimitTheorem 173

Problems 176

4 Parametric Point Estimation 177

4.1 Statistics 177

4.1.1 Sampling Distributions 178

4.1.2 Unbiased Statistics and Estimators 179

4.1.3 Standard Error and Mean Squared Error 181

4.1.4 The Delta Method 186

Problems 186

4.2 Sufficient Statistics 190

4.2.1 Exponential Family Distributions 195

Problems 200

4.3 Minimum Variance Unbiased Estimators 203

4.3.1 Cramér-Rao Lower Bound 205

Problems 212

4.4 Case Study -The Order Statistics 214

Problems 219

5 Likelihood-based Estimation 223

5.1 Maximum Likelihood Estimation 226

5.1.1 Properties of MLEs 226

5.1.2 One-parameter Probability Models 228

5.1.3 Multipara