# 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.