Multivariate Bonferroni-Type Inequalities: Theory and Applications

John Chen



Multivariate Bonferroni-Type Inequalities: Theory and Applications presents 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 Fréchet 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.

Features

• Covers theoretical and applied developments on inequality theory for future investigations
• Intertwines the latest bounding techniques with up-to-date applications in clinical trials, engineering, actuarial science, and other areas
• Emphasizes heuristic statistical thinking on approximating the probability of unions of several sets of events that occur simultaneously
• Explains how multivariate bounding is not a parallel extension of univariate bounding
• Goes beyond traditional Bonferroni bounds by presenting vectorized upper and hybrid bounds
• Devotes an entire chapter to case studies of applications of probability inequalities



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