Lecture Topics from Casella and Berger Book, Fall 2014 ======================================================. LECTURE 1. (9/3/14) Review. Brief general introduction: what is Statistics ? From Ch.1, mixed-type distributions (cf. Example 1.5.6 on CDFs with jumps). LECTURE 2. (9/8/14) Review. From Ch.2, functions of single random variables, transformation of densities (Thms 2.1.3, 2.1.5, 2.1.8 and Probability Integral Transform). Basic idea of stochastic simulation. LECTURE 3. (9/10/14) From Sec. 2.3: Moment generating functions, basic properties. Sec. 2.4: Differentiating under the integral sign. From Ch.3: Secs 3.4-3.5: Definitions and basic properties of exponential families and location-scale families. LECTURE 4. (9/15/14) Wrap-up from Ch.2. --- Using moments to characterize distributions (and sometimes prove limit theorems). Secs 3.2-3.3: Characterizing familiar distributions by qualitative properties. LECTURE 5. (9/17/14) From Ch.4, Secs.4-1-4.3 and Example 4.6.13: joint and conditional probability mass functions and densities; independence and dependence of jointly distributed r.v.'s. Jacobian change of variable formula for densities. LECTURE 6. (9/22/14) Mixed-type joint probability distributions. Conditional expectations and variances, repeated conditioning. Sec.4.4: Hierarchical and mixture models. LECTURE 7. (9/24/14) EXTRA topic: Multivariate normal distribution, alternative definitions. Covariance and dependence for normal r.v.'s. Independece of Xbar and S^2 for normal samples. LECTURE 8. (9/29/14) Ch.5: Random samples, sampling distributions. Distributions of sample sums and averages. Independence of sample means and variances of normals, and definition of t. LECTURE 9. (10/1/14) Ch.5, continued. Distributions of sample means and variances of normals: t, chi-square and F. Order-statistics: joint and marginal densities for continuously distributed samples. LECTURE 10. (10/6/14) Memoryless property and order statistics. Summary/Review/Overview in preparation for Test Wed. 10/8. LECTURE 11. (10/13/14) Ch.5, continued Discussion of Test Problem Solutions. Simulation methods: conditional distribution simulation, accept-reject methods. Lecture 12. (10/15/14) Ch.6. Sufficient Statistics, Motivation, Definition and Examples. Factorization Theorem. Proof in discrete cases and continuous cases in which smooth transformations (T,V) <-> X exist. Lecture 13. (10/20/14 Ch.6, continued. Definition of Minimal or Complete Sufficient statistics. Criterion for Minimality. Exponential family examples. Lecture 14. (10/22/14) Ch.6. Ancillary Statistics. Examples of Minimal, Complete and Incomplete Statistics. Basu's Theorem. Lecture 15. (10/27/14). Wrap-up of Chapters 5 and 6. For 5: conditional probability integral theorem, applied to simulation. For 6: Likelihood function, formal likelihood principle, equivariance. Lecture 16. (10/29/14) Ch.7: Methods of finding point estimates. Method of Moments, MLE, and Bayes estimators. Examples involving Exponential families. Consult additional reading material in Conjugate-priors handout. Lecture 17. (11/3/14) More on MLE examples, Bayes estimators, and conjugate priors in exponential families. Mean-squared error of estimators. Lecture 18. (11/5/14) Further working out of Bayes examples using conjugate priors within natural exponential families. Lecture 19. (11/10/14) Ch.7 Best unbiased estimators: Cramer-Rao bound. Multivariate and biased-estimator version. Necessary condition for equality. Lecture 20. (11/12/14) Ch.7, Rao-Blackwell Theorem, Lehmann-Scheffe' Theorem, MVUE's. Examples of working out Rao-Blackwellized estimators and UMVUE's. Lecture 21. (11/17/14) Ch.7: Decision theory: ideas for evaluating estimators. Loss functions, Bayes, minimax and admissible estimators. Extra material (ref. to Rohatgi or Bickel & Doksum or Shao books) Lecture 22. (11/19/14) More on decision theory. Further discussion and examples of Decision Theory and Loss Function optimality. Lecture 23. (11/24/14) Ch.8: Intro to Hypothesis Testing. Neyman-Pearson Lemma. Lecture 24. (11/26/14) More on Neyman-Pearson. UMP Tests. Definition of MLR and Karlin-Rubin Thm. Lecture 25. (12/1/14) Likelihood Ratio Tests. Bayesian hypothesis tests. Lecture 26. (12/8/14) Discussion of 2nd in-class test. Lecture 27. (12/10/14) Review of hypothesis testing material. Extra material (ref. to Bickel & Doksum or Shao books): Estimating Equations. This material will be deferred to the Spring Term. I will produce a handout on this. Material on Numerical algorithms for numerical likelihood maximization deferred to next term: Newton-Raphson, Fisher scoring, and EM algorithm.