## STA 200B Introduction to Mathematical Statistics I

**Units:** 4

**Format:**

Lecture: 3 hours

Discussion: 1 hour

**Catalog Description:**

Sampling, methods of estimation, bias-variance decomposition, sampling distributions, Fisher information, confidence intervals, and some elements of hypothesis testing.

**Prerequisite:** STA 200A; 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
- 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)
- Fisher information CR--lower bound efficiency (5 lect)
- 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 (proof involves orthogonal transformations; maybe avoid proof?)
- Confidence intervals and bounds; concept of a pivot; (3 lect)
- Some elements of hypothesis testing: (5 lect) basic concepts: critical regions, level, size, power function; one--sided and two--sided tests; p--value; NP--framework, maybe t--test and F--test Suggested additional material to be covered reading assignments will be taken from the following material: -- Concept of sufficiency, factorization theorem and Rao--Blackwell theorem, admissibility, EM-- algorithm -- Bayesian analysis of normal samples: normal--gamma conjugate prior -- Credible intervals

**Restrictions:
** No credit to students who have taken course 131B.

**Illustrative reading:**

M.H. DeGroot and M.J. Shervish: Probability and Statistics, Addison Wesley (latest edition) G.G. Roussas: An introduction to Probability Theory and Statistical Inference, Elsevier

**GE3:**

SE

**Potential Overlap:**

The course material for STA 200B is the same as for STA 131B with the exception that students in STA 200B are given additional advanced reading material and additional homework assignments. The midterm and final examinations will differ from those of 131B in that they will include material covered in the additional reading assignments.

**History:**

First offered Winter 2017.