| Description |
1 online resource (147 pages) |
| Series |
SpringerBriefs in Probability and Mathematical Statistics |
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Springer briefs in probability and mathematical statistics.
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| Contents |
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 |
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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 |
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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 |
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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 |
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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 |
| Summary |
This book provides a unified exposition of some fundamental theoretical problems in high-dimensional statistics. It specifically considers the canonical problems of detection and support estimation for sparse signals observed with noise. Novel phase-transition results are obtained for the signal support estimation problem under a variety of statistical risks. Based on a surprising connection to a concentration of maxima probabilistic phenomenon, the authors obtain a complete characterization of the exact support recovery problem for thresholding estimators under dependent errors |
| Notes |
A.1 Auxiliary Facts of Chi-Square Distributions |
| Bibliography |
Includes bibliographical references |
| Notes |
Print version record |
| Subject |
Maxima and minima.
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Phase transformations (Statistical physics)
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Máximos y mínimos
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Transformaciones de fase (Física estadística)
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Maxima and minima
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Phase transformations (Statistical physics)
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Màxims i mínims.
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Transformacions de fase (Física estadística)
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| Genre/Form |
Llibres electrònics.
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| Form |
Electronic book
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| Author |
Stoev, Stilian
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| ISBN |
9783030809645 |
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3030809641 |
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