Statistical Theory: A Concise Introduction

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