Unimodality, Convexity, and Applications

Description

In this book, the basic notions and tools of unimodality as they relate to probability and statistics are presented. In addition, many applications are covered; these include the use of unimodality to obtain monotonicity properties of power functions of multivariate tests, minimum volume confidence regions, and recurrence of symmetric random walks. The diversity of the applications will convince the reader that unimodality and convexity form an important tool in the hands of a researcher in probability and statistics Content: Properties of Univariate Unimodal Distributions -- A General Definition of Unimodality on the Line -- The Convex Structure of the Set of Unimodal Distributions -- Convolution Properties of Unimodal Distributions -- Strong Unimodality -- The Gauss Inequality and the Three-Sigma Rule for Unimodal Distributions -- The Mean-Median-Mode Inequality -- Concepts of Multivariate Unimodality -- Some Definitions of Unimodality for Distributions on R[superscript n] -- Logconcave Distributions -- Preservation of Unimodality Properties under Weak Limits -- The Choquet Version of Central Convex Unimodality -- Interrelationships among the Definitions -- Marginal Distributions -- Convolutions -- Some More Notions of Unimodality -- Schur Concavity and Related Concepts of Unimodality -- Generalized Unimodality Indexed by a Positive Parameter -- Classes of Concave Densities and Measures -- Unimodality for Discrete Distributions -- The Structure of Unimodal Discrete Distributions -- Convolutions of Discrete Unimodal Distributions and Strong Unimodality -- Unimodality of High Convolutions -- Unimodality of Infinitely Divisible Distributions -- Infinitely Divisible Distributions -- Unimodality of Symmetric Infinitely Divisible Distributions -- The Unimodality of all Distributions in L -- The Unimodality of Multivariate Infinitely Divisible Distributions -- Unimodality and Notions of Dependence -- Preliminaries on Notions of Dependence -- Monotonicity Properties of Probabilities Rectangular Regions -- Ordering of Distributions by Peakedness.