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UNC-Chapel Hill BIOS 761 - Advanced Probability and Statistical Inference II

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BIOS 761: Advanced Probability and Statistical Inference IIInstructor: Quefeng Li ([email protected])Office hours: 2:30 pm – 3:30 pm on Wednesday or by appointmentGrader: Yu Gu ([email protected]); Tianyou Luo ([email protected])Texts :• The course is taught based on self-contained class notes, which will be posted on Sakai.Please keep the notes to yourself.• Some useful reference books1. Mathematical Statistics: A Decision Theoretic Approach, by T. S. Ferguson, Pub-lisher: Academic Press.2. Statistical Decision Theory and Bayesian Analysis, by J. O. Berger, Publisher: Springer-Verlag.3. Mathematical Statistics, second edition, by J. Shao, Publisher: Springer-Verlag.4. Theory of Point Estimation, second edition, by E. L. Lehmann and G. Casella, Pub-lisher: Springer-Verlag.5. An Introduction to the Bootstrap, by B. Efron and R. Tibshirani, Publisher: Chap-man and Hall.6. The Jacknife and Bootstrap, by J. Shao and D. Tu, Publisher: Chapman and Hall.7. Elements of Statistical Learning, second edition, by T. Hastie, R. Tibshirani and J.Friedman, Publisher: Springer-Verlag.Prerequisite: BIOS 760Homework: There will be 6 homework assignments. Each problem is worth 10 points.Exams : There will be two in-class midterm exams and a final exam. The exam will not beaccumulative.Grading :1Homework 30%Exam 1: Chapter 1 (February 17) 20%Exam 2: Chapter 2 (March 18) 20%Final Exam: Chapter 3 & 4 (TBD) 30%Material to be covered1. Decision Theory (3 weeks)• Elementary decision theory, decision rules• Utility functions, loss functions, risk, minimax, and admissibility• Posterior distributions• Bayesian decision theory• Finding Bayes rules and Bayes risk• Finding minimax rules• Admissibility and inadmissibility2. Hypothesis Testing (4 weeks)• Likelihood ratio, Wald, and Score tests: Simple null and composite null hypothesis• Neyman-Pearson tests• Unbiased tests, UMP tests, UMPU tests, conditional tests– application to general exponential families• Bayesian hypothesis testing• Pitman efficiency• Pivotal quantities, confidence sets and their relationship to tests• Bayesian interval estimation, highest posterior density regions, credible sets3. Resampling Methods (3 weeks)• The Jackknife method• Bootstrap methods– Nonparametric bootstrap methods– Parametric bootstrap methods2– Bootstrap tests and confidence intervals• Cross-validation4. Topics in High-Dimensional Data Analysis (4 weeks)• High-dimensional penalized regression– Ridge estimator– LASSO in linear model and generalized linear model∗ Coordinate decent algorithm∗ Non-asymptotic theory– Other penalization methods: SCAD, adaptive LASSO, elastic net, Dantzig se-lector, fused LASSO– Variable selection with pre-defined structures– Penalization for other regression problems• Estimation of high-dimensional matrix– Estimation of sparse covariance matrix– Estimation of sparse precision matrix– Gaussian graphical model• High-dimensional classification– Linear and quadratic discriminant analysis– Support vector

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