Fundamentals of Mathematical Statistics

Fundamentals of Mathematical Statistics is meant for a standard one-semester advanced undergraduate or graduate-level course in Mathematical Statistics. It covers all the key topics—statistical models, linear normal models, exponential families, estimation, asymptotics of maximum likelihood, significance testing, and models for tables of counts. It assumes a good background in mathematical analysis, linear algebra, and probability but includes an appendix with basic results from these areas. Throughout the text, there are numerous examples and graduated exercises that illustrate the topics covered, rendering the book suitable for teaching or self-study. Features A concise yet rigorous introduction to a one-semester course in Mathematical Statistics Covers all the key topics Assumes a solid background in Mathematics and Probability Numerous examples illustrate the topics Many exercises enhance understanding of the material and enable course use This textbook will be a perfect fit for an advanced course in Mathematical Statistics or Statistical Theory. The concise and lucid approach means it could also serve as a good alternative, or supplement, to existing texts.

1. Statistical Models. 1.1. Models and parametrizations. 1.2. Likelihood, score, and information. 1.3. Exercises. 2. Linear Normal Models. 2.1. The multivariate normal distribution. 2.2. The normal distribution on a vector space. 2.3. The linear normal model. 2.4. Exercises. 3. Exponential Families. 3.1. Regular exponential families. 3.2. Examples of exponential families. 3.3. Properties of exponential families. 3.4. Constructing exponential families. 3.5. Moments, score, and information. 3.6. Curved exponential families. 3.7. Exercises. 4. Estimation. 4.1. General concepts and exact properties. 4.2. Various estimation methods. 4.3. The method of maximum likelihood. 4.4. Exercises. 5. Asymptotic Theory. 5.1. Asymptotic consistency and normality. 5.2. Asymptotics of moment estimators. 5.3. Asymptotics in regular exponential families. 5.4. Asymptotics in curved exponential families. 5.5. More about asymptotics. 5.6. Exercises. 6. Set Estimation. 6.1. Basic issues and definition. 6.2. Exact confidence regions by pivots. 6.3. Likelihood based regions. 6.4. Confidence regions by asymptotic pivots. 6.5. Properties of set estimators. 6.6. Credibility regions. 6.7. Exercises. 7. Significance Testing. 7.1. The problem. 7.2. Hypotheses and test statistics. 7.3. Significance and p-values. 7.4. Critical regions, power, and error types. 7.5. Set estimation and testing. 7.6. Test in linear normal models. 7.7. Determining p-values. 7.8. Exercises. 8. Models for Tables of Counts. 8.1. Multinomial exponential families. 8.2. Genetic equilibrium models. 8.3. Contingency tables. 8.4. Exercises.

Steffen Lauritzen is Emeritus Professor of Statistics at the University of Copenhagen and the University of Oxford as well as Honorary Professor at Aalborg University. He is most well known for his work on graphical models, in particular represented in a monograph from 1996 with that title, but he has published in a wide range of topics. He has received numerous awards and honours, including the Guy Medal in Silver from the Royal Statistical Society, where he also is an Honorary Fellow. He was elected to the Royal Danish Academy of Sciences and Letters in 2008 and became a Fellow of the Royal Society in 2011.