Lecture Topics from Casella and Berger Book, Fall 2014
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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.