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.
ntroduction 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.
