This page has been updated to reflect
topics actually covered for future reference.
Course Details:
Instructor: Todd Kuffner
Lecture: 8:30-10am, Monday and Wednesday, Cupples I, Room 199
Required textbooks:
- (AL): Athreya &
Lahiri's Measure Theory and
Probability Theory, First Edition, Springer.
- (TPE): Lehmann &
Casella's Theory of Point Estimation,
Second Edition, Springer.
- (TSH): Lehmann &
Romano's Testing Statistical
Hypotheses, Third Edition, Springer.
We also made use of notes (with generous permission) by Professor
Lester Mackey for Stats 300A at Stanford.
Exams: 1 midterm and 1 final
Homework: there were 7 homework assignments
Grades: 35% Homework, 30% Midterm, 35% Final
Topics List from probability (following
AL) (reflecting what was actually covered):
- Measures: introduction, extension theorems, Lebesgue-Stieltjes,
completeness
- Integration: measurable transformations, Fubini's theorem,
Riemann and Lebesgue integration
- General topology: L^{p} spaces, Banach and Hilbert spaces
- Differentiation: Radon-Nikodym theorem, signed measures
- Product Measures
- Probability spaces: random variables and random vectors
- Independence: Borel-Cantelli lemmas
Topics List from statistics (following
TPE
and
TSH) (reflecting what was
actually covered):
- Preliminaries: exponential families, group transformation models,
concentration inequalities
- Decision theory: loss, risk functions, admissibility, optimality
concepts such as unbiasedness, equivariance, minimum average risk
(Bayesian), minimax procedures
- Data reduction: Neyman-Fisher factorization, optimal data
reduction via minimal sufficiency, completeness, ancillarity, Basu's
theorem
- Risk reduction: convexity, Rao-Blackwell theorem, unbiasedness,
UMRUE, UMVUE, Lehmann-Scheffe theorem, minimum risk equivariant
estimation, the Pitman estimator, risk unbiasedness
- Bayes estimators: priors, admissibility, worst case optimality,
least favorable priors
- Simultaneous estimation: James-Stein estimator
- Hypothesis testing: Neyman-Pearson lemma, likelihood ratio test,
(uniformly) most powerful tests, monotone likelihood ratio,
UMPU tests, alpha-similar tests, alpha-Neyman structure, dealing with
nuisance parameters, UMP invariant tests, maximal invariants,
(uniformly most accurate) confidence regions