Mathematical statistics
Graduate · Statistics
Syllabus focus
Theoretical / proof-based
Pricing
Graduate-level rates are set on consultation. See the pricing page for K–12 and undergraduate rates.
Topics typically covered
Theoretical / proof-based
Probability foundations
- Probability spaces, sigma-algebras, and measures
- Random variables and induced measures
- Expectation via Lebesgue integral
- Independence and product measures
- Convergence modes: a.s., in probability, Lp, in distribution
Distribution and limit theory
- Characteristic functions
- Law of large numbers and central limit theorems
- Multivariate normal and quadratic forms
- Sufficient, complete, and ancillary statistics
- Exponential families
Statistical inference theory
- Point estimation: UMVU, MLE, and efficiency
- Hypothesis testing: Neyman–Pearson and UMP tests
- Confidence sets and duality with testing
- Asymptotic theory: delta method and Fisher information
- Decision theory and admissibility (introduction)
Notes
First core course in many statistics PhD programs. Expect proof-based treatment aligned with Casella & Berger, Bickel & Doksum, or similar texts.