AMS 571: Mathematical Statistics

Sampling distribution; convergence concepts; classes of statistical models; sufficient statistics; likelihood principle; point estimation; Bayes estimators; consistency; Neyman-Pearson Lemma; UMP tests; UMPU tests; Likelihood ratio tests; large sample theory.

Required Textbook: Statistical Inference by George Casella and Roger L. Berger, 2nd edition, 2002, Duxbury Advanced Series

Supplementary Textbooks:

1. Introduction to Mathematical Statistics by Robert Hogg, Joseph McKean and Allen Craig, 8th edition, 2018, Pearson
2. Mathematical Statistics and Data Analysis by John A. Rice, 3rd edition, 2006, Cengage
3. Introduction to Mathematical Statistics and Its Applications by Richard Larsen and Morris Marx, 6th edition, 2017, Pearson
4. John E. Freund’s Mathematical Statistics with Applications by Irwin Miller and Marylees Miller, 8th edition, 2018, Pearson
5. Theory of Point Estimation by Erich L. Lehmann and George Casella, 2nd edition, 1998, Springer
6. Theoretical Statistics: Topics for a Core Course by Robert W. Keener, 2010, Springer

Learning Outcomes

1. Demonstrate deep understanding of mathematical concepts on statistical methods in:
   • Sampling and large-sample theory;
   • Sufficient, ancillary and complete statistics;
   • Point estimation;
   • Hypothesis testing;
   • Confidence interval.
2. Demonstrate deep understanding in advanced statistical methods including:
   • Maximum likelihood, method of moment and Bayesian methods;
   • Evaluation of point estimators, mean squared error and best unbiased estimator;
   • Evaluation of statistical tests, power function and uniformly most powerful test;
   • Interval estimation based on pivot quantity or inverting a test statistic.
3. Demonstrate skills with solution methods for theoretical proofs:
   • Almost sure convergence, convergence in probability and convergence in distribution;
   • Ability to follow, construct, and write mathematical/statistical proofs;
   • Ability to derive theoretical formulas for statistical inference in real-world problems.
4. Develop proper skillsets to conduct statistical research:
   • Ability to understand and write statistical journal papers;
   • Ability to develop and evaluate new statistical methods;
   • Ability to adopt proper statistical theories in research.