A Concise Introduction
Chapman and Hall/CRC – 2013 – 240 pages
Designed for a one-semester advanced undergraduate or graduate course, Statistical Theory: A Concise Introduction clearly explains the underlying ideas and principles of major statistical concepts, including parameter estimation, confidence intervals, hypothesis testing, asymptotic analysis, Bayesian inference, and elements of decision theory. It introduces these topics on a clear intuitive level using illustrative examples in addition to the formal definitions, theorems, and proofs.
Based on the authors’ lecture notes, this student-oriented, self-contained book maintains a proper balance between the clarity and rigor of exposition. In a few cases, the authors present a "sketched" version of a proof, explaining its main ideas rather than giving detailed technical mathematical and probabilistic arguments. Chapters and sections marked by asterisks contain more advanced topics and may be omitted. A special chapter on linear models shows how the main theoretical concepts can be applied to the well-known and frequently used statistical tool of linear regression.
Requiring no heavy calculus, simple questions throughout the text help students check their understanding of the material. Each chapter also includes a set of exercises that range in level of difficulty.
"As teachers of theoretical statistics, we can use a new approach, which this text offers. … a helpful resource for teachers of mathematical statistics who are looking for an outline of teaching material and useable depth. Their material attains a workable syllabus, which can be easily augmented with the teacher’s preferred emphasis. This volume will make a solid contribution to any theoretical statistics instructor’s collection due to its convenient size, its scope of coverage, judicious use of examples, and clarity of exposition."
—The American Statistician, May 2014
Exponential family of distributions
Maximum likelihood estimation
Method of moments
Method of least squares
Goodness-of-estimation. Mean squared error.
Confidence Intervals, Bounds, and Regions
Quoting the estimation error
Hypothesis testing and confidence intervals
Convergence and consistency in MSE
Convergence and consistency in probability
Convergence in distribution
The central limit theorem
Asymptotically normal consistency
Asymptotic confidence intervals
Asymptotic normality of the MLE
Asymptotic distribution of the GLRT. Wilks’ theorem.
Choice of priors
Interval estimation. Credible sets.
Elements of Statistical Decision Theory
Introduction and notations
Risk function and admissibility
Minimax risk and minimax rules
Bayes risk and Bayes rules
Posterior expected loss and Bayes actions
Admissibility and minimaxity of Bayes rules
Definition and examples
Estimation of regression coefficients
Residuals. Estimation of the variance.
Goodness-of-fit. Multiple correlation coefficient.
Confidence intervals and regions for the coefficients
Hypothesis testing in linear models
Analysis of variance
Appendix A: Probabilistic Review
Appendix B: Solutions of Selected Exercises
Exercises appear at the end of each chapter.