Mathematical statistics : an introduction to likelihood based inference

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. Contents......Page 3 Preface......Page 8 -Algebras......Page 11 Problems......Page 17 Probability Axioms and Rules......Page 19 Problems......Page 24 Probability with Equally Likely Outcomes......Page 26 Problems......Page 28 Conditional Probability......Page 29 Problems......Page 35 Independence......Page 38 Problems......Page 41 Counting Methods......Page 43 Problems......Page 48 Case Study – The Birthday Problem......Page 51 Problems......Page 54 Random Variables......Page 55 Problems......Page 60 Random Vectors......Page 63 Problems......Page 70 Independent Random Variables......Page 73 Problems......Page 76 Transformations of Random Variables......Page 78 Problems......Page 85 Expected Values for Random Variables......Page 87 Problems......Page 99 Expected Values for Random Vectors......Page 104 Problems......Page 120 Sums of Random Variables......Page 124 Problems......Page 130 Case Study – How Many Times was the Coin Tossed?......Page 133 Problems......Page 136 Discrete Probability Models......Page 138 Problems......Page 149 Continuous Probability Models......Page 156 Problems......Page 167 Important Distributional Relationships......Page 172 Problems......Page 179 Case Study – The Central Limit Theorem......Page 181 Problems......Page 185 Statistics......Page 186 Problems......Page 195 Sufficient Statistics......Page 199 Problems......Page 209 Minimum Variance Unbiased Estimators......Page 212 Problems......Page 221 Case Study – The Order Statistics......Page 223 Problems......Page 228 Likelihood-based Estimation......Page 231 Maximum Likelihood Estimation......Page 234 Problems......Page 248 Bayesian Estimation......Page 255 Problems......Page 263 Interval Estimation......Page 266 Problems......Page 277 Case Study – Modeling Obsidian Rind Thicknesses......Page 281 Problems......Page 286 Hypothesis Testing......Page 288 Components of a Hypothesis Test......Page 289 Problems......Page 293 Most Powerful Tests......Page 295 Problems......Page 300 Uniformly Most Powerful Tests......Page 303 Problems......Page 308 Generalized Likelihood Ratio Tests......Page 312 Problems......Page 318 Large Sample Tests......Page 321 Problems......Page 327 Case Study – Modeling Survival of the Titanic Passengers......Page 330 Problems......Page 335 Generalized Linear Models......Page 337 Generalized Linear Models......Page 338 Problems......Page 340 Fitting a Generalized Linear Model......Page 342 Problems......Page 346 Hypothesis Testing in a Generalized Linear Model......Page 347 Problems......Page 352 Generalized Linear Models for a Normal Response Variable......Page 354 Problems......Page 368 Generalized Linear Models for a Binomial Response Variable......Page 371 Problems......Page 379 Case Study – IDNAP Experiment with Poisson Count Data......Page 381 Problems......Page 387 Refs......Page 389 Probability Models......Page 391 Data Sets......Page 393 Solutions for Chapter 1......Page 395 Solutions for Chapter 2......Page 398 Solutions for Chapter 3......Page 403 Solutions for Chapter 4......Page 406 Solutions for Chapter 5......Page 408 Solutions for Chapter 6......Page 411 Solutions for Chapter 7......Page 415 Index......Page 418 Explores mathematical statistics in its entirety, from the fundamentals to modern methods This book introduces readers to point estimation, confidence intervals, and statistical tests. Based on the general theory of linear models, it provides an in-depth overview of the following: analysis of variance (ANOVA) for models with fixed, random, and mixed effects; regression analysis is also first presented for linear models with fixed, random, and mixed effects before being expanded to nonlinear models; statistical multi-decision problems like statistical selection procedures (Bechhofer and Gupta) and sequential tests; and design of experiments from a mathematical-statistical point of view. Most analysis methods have been supplemented by formulae for minimal sample sizes. The chapters also contain exercises with hints for solutions. Translated from the successful German text, Mathematical Statistics requires knowledge of probability theory (combinatorics, probability distributions, functions and sequences of random variables), which is typically taught in the earlier semesters of scientific and mathematical study courses. It teaches readers all about statistical analysis and covers the design of experiments. The book also describes optimal allocation in the chapters on regression analysis. Additionally, it features a chapter devoted solely to experimental designs.-Classroom-tested with exercises included -Practice-oriented (taken from day-to-day statistical work of the authors) -Includes further studies including design of experiments and sample sizing -Presents and uses IBM SPSS Statistics 24 for practical calculations of data Mathematical Statistics is a recommended text for advanced students and practitioners of math, probability, and statistics. '