Statistical Theory: A Concise Introduction
Features
- Delineates the fundamental concepts of statistical theory
- Balances exposition and mathematical formality
- Introduces topics with numerous examples, avoiding a dry approach to the subject
- Assumes an intermediate-level background in calculus and probability
- Reviews the necessary probabilistic material in an appendix
- Includes a series of exercises of varying levels of difficulty in each chapter, with selected solutions in an appendix
Summary
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.
Table of Contents
Introduction
Preamble
Likelihood
Sufficiency
Minimal sufficiency
Completeness
Exponential family of distributions
Point Estimation
Introduction
Maximum likelihood estimation
Method of moments
Method of least squares
Goodness-of-estimation. Mean squared error.
Unbiased estimation
Confidence Intervals, Bounds, and Regions
Introduction
Quoting the estimation error
Confidence intervals
Confidence bounds
Confidence regions
Hypothesis Testing
Introduction
Simple hypotheses
Composite hypotheses
Hypothesis testing and confidence intervals
Sequential testing
Asymptotic Analysis
Introduction
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
Multiparameter case
Asymptotic distribution of the GLRT. Wilks’ theorem.
Bayesian Inference
Introduction
Choice of priors
Point estimation
Interval estimation. Credible sets.
Hypothesis testing
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
Linear Models
Introduction
Definition and examples
Estimation of regression coefficients
Residuals. Estimation of the variance.
Examples
Goodness-of-fit. Multiple correlation coefficient.
Confidence intervals and regions for the coefficients
Hypothesis testing in linear models
Predictions
Analysis of variance
Appendix A: Probabilistic Review
Appendix B: Solutions of Selected Exercises
Index
Exercises appear at the end of each chapter.
