Theory of Statistics III

The main goal of this course is to expose students to modern ideas in statistical theory to bring them to the frontier of research. We will study: (1) testing problems in high dimensions understanding the performance of Bonferroni's method, Fisher's test, chi-square tests, and the higher criticism under sparse alternatives with strong effects and denser alternatives with mild effects; (2) multiple testing problems, the familywise error rate (FWER) and procedures for controlling the FWER, false discovery rate (FDR), Benjamini-Hochberg procedure; (3) conditional testing and controlled variable selection via knockoffs; (4) combining results from several tests via e-values and anytime valid inference; (5) topics in selective inference such as false coverage rate and post-selection inference; (6) conformal/predictive inference; (7) permutation testing and its modern applications in model-free inference ; (8) James-Stein estimation; (9) empirical Bayes methods.