Instructors: Harrison H. Zhou .
Email: huibin.zhou@yale.edu
TA: Peisi Yan
Email: peisi.yan@yale.edu
Class Time and Place: M&W 2:30-3:45pm in Room 107, 24 Hillhouse Ave
Course Description: Shrinkage estimation and its connection to minimaxity, admissibility, Bayes, empirical Bayes, and hierarchical Bayes. Shrinkage captures essential nonlinearity necessary to outperform standard linear estimators in Gaussian regression models and random effects models. Relationship to model selection and to sparsity in the estimation of functions by selection from large dictionaries of candidate terms. Nonparametric estimation. Tests of statistical hypotheses. Multiple comparisons. Some knowledge of statistical theory at the level of STAT 610a is assumed.
References:
Lecture notes of Lawrence D. Brown, Shrinkage: Fall 2006
David B. Pollard. Asymptopia
Iain Johnstone. "Function
estimation and Gaussian sequence model" . Papers: Wald: 1939. Brown: 1971; Brown and
Hwang: 1982;
Berger and Srinivasan: 1978. Robbins (1951, 1956), Efron: 2003. Seeger: 1968. Abramovich, Benjamini,
Donoho and Johnstone: 2006. Grade: Participation: 25% Tentative Lectures: Week 1: pdf. Week 2: pdf. Week 3: pdf. Week 4: pdf. Week 5: pdf. Week 6: pdf. Week 7: pdf. Week 8: pdf. Week 9: pdf. Week 10: pdf. Week 11: pdf. Week 12: pdf. Week 13: pdf. Tentative Topics: Topic 1. Shrinkage estimation in
parametric models. 4-6 weeks.
Homework every other week: 75%
(ii) Bayes estimation, minimaxity and Admissibility.
(iii) Empirical Bayes, hierarchical Bayesand random effects.
Topic 2. Shrinkage estimation in nonparametric models. 1-2 weeks.
(i) Best linear estimation.
(ii) Blockwise Stein's estimation and sharp adaptive minimaxity.
Topic 3. Testing hypothesis and minimax estimation. 5-6 weeks.
(i) Neyman-Pearson Lemma and minimax lower bound.
(ii) Minimax estimation for functional data analysis.
(iii) Minimax Estimation for covariance matrices estimation.
(iv) Muitiple comparisons and sharp adaptive minimaxity.
Topic 4. Le Cam Theory. 0-2 weeks.
(i) Asymptotic equivalence theory.