Texts: required Peter Bickel and Kjell Doksum, Mathematical
Statistics, vol.I, 2nd ed., Pearson Prentice Hall, 2007.
(recommended)
V. Rohatgi and A.K. Saleh, An Introduction to Probability and Statistics,
2nd ed., Wiley.
I. Background & Review on Sampling Distributions (2.5 weeks)
A. Moment generating functions, Conditioning, Transformation of random vectors
(Bickel & Doksum Appendix A.12, B.1-B.2, Rohatgi & Saleh 3.3, 4.4, 4.6)
B. Multivariate Normal Distribution (Bickel & Doksum B.4, B.6; Rohatgi & Saleh 5.4)
C. Distribution Theory for Normal Samples -- Chi-square, t, F
(Bickel & Doksum B.3, Rohatgi & Saleh Ch.7)
II. Statistical Models, Goals & Performance Criteria (B & D Ch.1 through 1.4) (2.5 weeks)
A. Data Structures, Models, Parameter spaces
B. Decision Theory
C. Sufficiency (with supplementary material from Rohatgi & Saleh 8.3)
III. Exponential Families -- Natural & Curved (B & D 1.6) (2 weeks)
A. Completeness. Generalized Moment Estimators.
B. Conjugate families of Priors
IV. Principles of Estimation: Minimum Contrast, ML, Method of Moments (B & D Ch.2) (2 weeks)
A. Definitions & basic properties. Least squares. M-estimation.
B. Algorithms including Hill-Climbing & EM
V. Performance Criteria for Statistical Procedures (B&D Ch.3 plus R&S 8.4-8.5) (2 weeks)
A. Bayes & Minimax
B. Information & Minimum Variance Unbiased Estimation
VI. Optimal Hypothesis Testing and Confidence Regions (B&D Ch.4) (3 weeks)
A. Neyman Pearson Lemma, UMP Tests, Monotone Likelihood Ratio
B. Confidence regions vs Hypothesis Testing.
C (Generalized) Likelihood Ratio Tests.