Statistical inference
Undergraduate · Statistics
Syllabus focus
Standard syllabus · Theoretical / proof-based
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Topics typically covered
Standard syllabus
Estimation
- Point estimators: method of moments and maximum likelihood (intro)
- Properties of estimators: bias, variance, and MSE
- Confidence intervals: pivot methods
- Large-sample confidence intervals
- Bootstrap confidence intervals (introduction)
Hypothesis testing
- Null and alternative hypotheses; test statistics
- P-values and significance levels
- Tests for means, proportions, and variances
- Likelihood ratio tests (introduction)
- Multiple testing overview: family-wise error
- Power and sample size (introduction)
Advanced inference topics
- Sufficient statistics and the factorization theorem (statement)
- Rao–Blackwell and Cramér–Rao lower bound (statement)
- Bayesian credible intervals (preview)
- Nonparametric tests preview: sign and Wilcoxon
Theoretical / proof-based
Decision-theoretic foundations
- Loss functions and risk
- Completeness and sufficiency with proofs
- Exponential families and canonical forms
- UMVU estimators via Rao–Blackwell
- Cramér–Rao inequality: proof and applications
Asymptotic theory
- Consistency and asymptotic normality of MLE
- Delta method
- Asymptotic efficiency
- Likelihood ratio test: Wilks' theorem (statement)
- Uniformly most powerful tests (introduction)
Notes
Often follows probability in the statistics major sequence. Theoretical sections emphasize derivations and asymptotics; standard sections focus on methods and interpretation.