Instructors: Harrison H. Zhou and Andrew R. Barron
e-mails: huibin.zhou@yale.edu for
Harrison Zhou and andrew.barron@yale.edu
for Andrew Barron.
Class Time: Tuesday and Thursday 4:00-5:15pm.
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% 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: Leung
and Barron (2004) Week 10: pdf Week 11: pdf. Week 12: pdf. Week 13: pdf. Tentative schedule: Topic 1. Shrinkage estimation in
parametric models (6 weeks) i. The Canonical normal means estimation problem. Stein's unbiased estimator of risk. ii. Bayes estimation, minimaxity and
Admissibility. Complete class theorem. iii.
Empirical Bayes, hierarchical Bayes
and random effects. iv.
Shrinkage estimation in the absence of spherical symmetry. Topic 2. Shrinkage
estimation in nonparametric models (2 weeks, Pinsker
bound theory) i. Best linear estimation. ii. Blockwise Stein's estimation
and Adaptive minimaxity. Minimax
Theorem. Topic 3. Testing
hypothesis and its connection to estimation. (3 weeks). i. Neyman-Pearson Lemma. Connection to minimax lower bound. ii.
Multiple comparison and false discovery rate (FDR). Connection
to model selection and sparse estimation. Topic 4. Le Cam theory. (2 weeks) i. Hellinger differentiability and asymptotic normality. ii. Asymptotic equivalence theory
Weekly Homework: 75%