Multivariate Bonferroni-Type Inequalitiespresents a systematic account of research discoveries on multivariate Bonferroni-type inequalities published in the past decade. The emergence of new bounding approaches pushes the conventional definitions of optimal inequalities and demands new insights into linear and Frechet optimality. The book explores these advances in bounding techniques with corresponding innovative applications. It presents the method of linear programming for multivariate bounds, multivariate hybrid bounds, sub-Markovian bounds, and bounds using Hamilton circuits. The first half of the book describes basic concepts and methods in probability inequalities. The author introduces the classification of univariate and multivariate bounds with optimality, discusses multivariate bounds using indicator functions, and explores linear programming for bivariate upper and lower bounds. The second half addresses bounding results and applications of multivariate Bonferroni-type inequalities.The book shows how to construct new multiple testing procedures with probability upper bounds and goes beyond bivariate upper bounds by considering vectorized upper and hybrid bounds. It presents an optimization algorithm for bivariate and multivariate lower bounds and covers vectorized high-dimensional lower bounds with refinements, such as Hamilton-type circuits and sub-Markovian events. The book concludes with applications of probability inequalities in molecular cancer therapy, big data analysis, and more.
List of Figures xi
List of Tables xiii
Preface xv
1 Introduction 1 (26)
1.1 Multiple Extreme Values 2 (4)
1.1.1 Evaluating the Risk of Multiple 2 (3)
Disasters
1.1.2 Multivariate Cumulative Distributions 5 (1)
1.2 Minimum Effective Dose 6 (7)
1.2.1 Minimum Effective Dose of MOTRIN 6 (4)
1.2.2 Minimum Effective Dose without 10 (1)
Normality
1.2.3 Inequality Methods for Behren-Fisher 11 (1)
Problem
1.2.4 Adjusting Multiplicity for Two or 12 (1)
More Substitutable Endpoints
1.3 System Reliability 13 (5)
1.3.1 Basic Systems 13 (2)
1.3.2 Composite Systems 15 (3)
1.4 Education Reform and Theoretical Windows 18 (4)
1.4.1 Learning Outcomes of Different 19 (1)
Pedagogies
1.4.2 Therapeutic Windows of a Drug 20 (2)
1.5 Ruin Probability and Multiple Premiums 22 (2)
1.6 Martingale Inequality and Asset Portfolio 24 (3)
2 Fundamentals 27 (56)
2.1 Univariate Bonferroni-type Bounds 30 (11)
2.1.1 Linear Combination Bounds 30 (5)
2.1.2 Non-linear Combination Bounds 35 (6)
2.2 Univariate Optimality 41 (18)
2.3 Multivariate Bounds 59 (15)
2.3.1 Complete Bonferroni Summations 59 (2)
2.3.2 Partial Bonferroni Summations 61 (3)
2.3.3 Decomposition of Bonferroni Summations 64 (3)
2.3.4 Classical Bonferroni Bounds in a 67 (7)
Multivariate Setting
2.4 Multivariate Optimality 74 (9)
3 Multivariate Indicator Functions 83 (28)
3.1 Method of Indicator Functions 83 (6)
3.2 Moments of Bivariate Indicator Functions 89 (10)
3.2.1 Bounds for Joint Probability of 89 (6)
Exactly r Occurrences
3.2.2 Bounds for Joint Probability of at 95 (4)
Least r Occurrences
3.3 Factorization of Indicator Functions 99 (6)
3.4 A Paradox on Factorization and Binomial 105 (6)
Moments
3.4.1 Upper Bound Inconsistency 105 (5)
3.4.2 Lower Bound Inconsistency 110 (1)
4 Multivariate Linear Programming Framework 111 (32)
4.1 Linear Programming Upper Bounds 112 (11)
4.1.1 Matrix Expression of Upper Frechet 113 (1)
Optimality
4.1.2 Target Function of Linear Programming 114 (2)
4.1.3 Linear Programming Constraints 116 (3)
4.1.4 Duality Theorem and Existence of 119 (4)
Optimality
4.2 Linear Programming Lower Bounds 123 (20)
4.2.1 Inconsistency of Linear Programming 124 (5)
Lower Bounds
4.2.2 Feasible Linear Programming Lower 129 (2)
Bounds
4.2.3 A Perturbation Device in Linear 131 (3)
Programming Optimization
4.2.4 An Iteration Process in Linear 134 (9)
Programming Optimization
5 Bivariate Upper Bounds 143 (24)
5.1 Bivariate Factorized Upper Bounds 143 (4)
5.2 Bivariate High-degree Upper Bounds 147 (3)
5.3 Bivariate Optimal Upper Bounds 150 (10)
5.3.1 Linear Optimal Upper Bounds 150 (5)
5.3.2 Bivariate Frechet Optimal Upper Bounds 155 (5)
5.4 Applications in Multiple Testing 160 (7)
5.4.1 Bonferroni Procedure 161 (1)
5.4.2 Holm Step-down Procedure 162 (1)
5.4.3 Improved Holm Procedure 163 (4)
6 Multivariate and Hybrid Upper Bounds 167 (26)
6.1 High Dimension Upper Bounds 167 (10)
6.2 Hybrid Upper Bounds 177 (7)
6.3 Applications in Successive Comparisons 184 (9)
6.3.1 Equal Variances 186 (1)
6.3.2 Unequal Variances, Behrens-Fisher 186 (7)
Problem
7 Bivariate Lower Bounds 193 (30)
7.1 Bivariate Factorized Lower Bounds 193 (10)
7.2 Bivariate High-degree Lower Bounds 203 (1)
7.3 Bivariate Optimal Factorized Bounds 204 (4)
7.4 Bivariate Optimal Algorithm Bounds 208 (13)
7.5 Applications in Seasonal Trend Analysis 221 (2)
8 Multivariate and Hybrid Lower Bounds 223 (28)
8.1 High Dimension Lower Bounds 223 (5)
8.2 Hybrid Lower Bounds 228 (21)
8.2.1 Setting of Hybrid Lower Bounds 229 (5)
8.2.2 Main Results of Hybrid Lower Bounds 234 (10)
8.2.3 Examples of Hybrid Lower Bounds 244 (5)
8.3 Applications in Outlier Detection 249 (2)
9 Case Studies 251 (18)
9.1 Molecular Cancer Therapy 251 (2)
9.2 Therapeutic Window 253 (3)
9.3 Minimum Effective Dose with 256 (2)
Heteroscedasticity
9.4 Simultaneous Inference with Binary Data 258 (3)
9.5 Post-thrombotic Syndrome and Rang 261 (3)
Regression
9.6 Vascular Risk Assessment 264 (1)
9.7 Big-data Analysis 265 (4)
Bibliography 269 (14)
Index 283
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