Objectives and Overview

This coursework page combines materials and notes for the core graduate mathematical statistics sequence at USC:

• Math 541A: Graduate Mathematical Statistics I (Spring 2024)

• Math 541B: Graduate Mathematical Statistics II (Fall 2024) and core graduate mathematical statistics courses at Yonsei University:

• STA6010: Graduate Mathematical Statistics I (Spring 2022)

Topics

1. Probability Foundations: Probability spaces, random variables, expectations, inequalities, convergence modes.
2. Distribution Theory: Transformations, families of distributions (exponential families, location-scale families), sufficient statistics, ancillary statistics, complete statistics.
3. Point Estimation: Methods of estimation (Method of moments, Maximum likelihood), criteria for evaluating estimators (MSE, UMVUE).

4. Information Inequality: Cramér-Rao lower bound, Fisher Information.

5. Hypothesis Testing: Neyman-Pearson Lemma, Uniformly Most Powerful (UMP) tests, Likelihood ratio tests.
6. Confidence Intervals: Inverting test statistics, pivotal quantities.
7. Asymptotic Theory: Delta method, asymptotic relative efficiency, asymptotics of MLEs and likelihood ratio tests.
8. Advanced Topics: Bayesian analysis basics, minimax estimation, and nonparametric methods.

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