STA 131B Introduction to Mathematical Statistics
Lecture: 3 hours
Discussion: 1 hour
Sampling, methods of estimation, bias-variance decomposition, sampling distributions, Fisher information, confidence intervals, and some elements of hypothesis testing.
Prerequisite: STA 131A; or Consent of Instructor.
Summary of course contents:
- Intro (2 lect.): Concept of a statistical model; observations as random variables, definition/examples of a statistic, statistical inference and examples throughout the entire course: emphasize the difference between population quantities, random variables and observables
- Methods of estimation: MLEs, Bayes, MOM (5 lect.) including: (a) likelihood function; finding MLEs (finding a global maximum of a function) invariance of MLE; some limitations of ML-approach; exponential families; (b) Bayes approach, loss/risk functions; conjugate priors
- MSE; bias-variance decomposition, unbiased estimation (2 lect) (IV) Sampling distributions: (5 lect) (a) distributions of transformed random variables; (b) t, F and chi^2 (properties:mgf, pdf, moments...); (c) sampling distribution of sample variance under normality; independence of sample mean and sample variance under normality (V) Fisher information CR-lower bound efficiency (5 lect)
- Confidence intervals and bounds; concept of a pivot; (3 lect)
- Some elements of hypothesis testing: (5 lect) critical regions, level, size, power function, one-sided and two-sided tests; p-value); NP-framework, perhaps t-test
Probability and Statistics by Mark J. Schervish, Morris H. DeGroot 4th Edition 2014, Pearson