Intro
Preface
Contents
Acronyms
1 Introduction and Guiding Examples
1.1 The Additive Error Model
1.2 Genome-Wide Association Studies and the Chi-Square Model
1.3 Contents
2 Risks, Procedures, and Error Models
2.1 Statistical Risks
2.2 Statistical Procedures
2.3 Related Literature and Our Contributions
2.4 Relationships Between the Asymptotic Risks
2.5 The Asymptotic Generalized Gaussian (AGG) Models
2.6 Rapid Variation and Relative Stability
2.7 Auxiliary Facts About Gaussian Distributions
3 A Panorama of Phase Transitions

3.1 Sparse Signal Detection Problems
3.2 Sparse Signal Support Recovery Problems
3.3 The Exact Support Recovery Problem
3.4 The Approximate Support Recovery Problem
3.5 Monotonicity of the Benjamini-Hochberg Procedure
3.6 The Exact-Approximate Support Recovery Problem
3.7 The Approximate-Exact Support Recovery Problem
3.8 Asymptotic Power Analysis: A Discussion
4 Exact Support Recovery Under Dependence
4.1 Generalizations of Distributional and Dependence Assumptions
4.2 Sufficient Conditions for Exact Support Recovery
4.3 Dependence and Uniform Relative Stability

4.4 Necessary Conditions for Exact Support Recovery
4.5 Dense Signals
4.6 Numerical Illustrations for Independent Errors
5 Bayes and Minimax Optimality
5.1 Bayes Optimality in Support Recovery Problems
5.2 Bayes Optimality of Oracle Thresholding
5.3 Bayes Optimality of Likelihood Ratio Thresholding
5.4 Sub-optimality of Data Thresholding Procedures
5.5 Minimax Optimality in Exact Support Recovery
5.5.1 Point-Wise Minimax Optimality for Thresholding Procedures
5.5.2 Minimax Optimality over All Procedures
5.6 Optimality and Sub-optimality: A Discussion

6 Uniform Relative Stability for Gaussian Arrays
6.1 Ramsey's Theory and the Structure of Correlation Matrices
6.2 URS Implies UDD (Proof of the ``Only If'' Part of Theorem6.1)
6.3 UDD Implies URS (Proof of the `If' Part of Theorem6.1)
6.3.1 Bounding the Upper Tails of AGG Maxima
6.3.2 Bounding the Lower Tails of Gaussian Maxima
6.4 Numerical Illustrations of Exact Support Recovery Under Dependence
7 Fundamental Statistical Limits in Genome-Wide Association Studies
7.1 Support Recovery Problems in Chi-Squared Models
7.1.1 The Exact Support Recovery Problem

7.1.2 The Exact-Approximate Support Recovery Problem
7.1.3 The Approximate Support Recovery Problem
7.1.4 The Approximate-Exact Support Recovery Problem
7.1.5 Comparison of One- Versus Two-Sided Alternatives in Additive Error Models
7.2 Odds Ratios and Statistical Power
7.3 Optimal Study Designs and Rare Variants
7.4 Phase Transitions in Large-Scale Association Screening Studies
7.5 Numerical Illustrations of the Phase Transitions in Chi-Square Models
7.5.1 Exact Support Recovery
7.5.2 Approximate, and Approximate-Exact Support Recovery
Appendix A Additional Proofs