Unimodality, convexity, and applications /
Sudhakar Dharmadhikari, Kumar Joag-dev.
Boston : Academic Press, c1988.
xiii, 278 p. : ill. --
More Details
added author
Boston : Academic Press, c1988.
general note
Includes indexes.
catalogue key
Bibliography: p. 261-269.
A Look Inside
About the Author
Author Affiliation
Sudhakar Dharmadhikari: Department of Mathematics, Southern Illinois University at Carbondale Kumar Joag-dev: Department of Mathematics, University of Illinois, Urbana
This item was reviewed in:
SciTech Book News, December 1988
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Main 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.
Table of Contents
Prefacep. xi
Properties of Univariate Unimodal Distributions
Summaryp. 1
A General Definition of Unimodality on the Linep. 1
The Convex Structure of the Set of Unimodal Distributionsp. 4
Convolution Properties of Unimodal Distributionsp. 11
Strong Unimodalityp. 17
The Gauss Inequality and the Three-Sigma Rule for Unimodal Distributionsp. 23
The Mean-Median-Mode Inequalityp. 33
Concepts of Multivariate Unimodality
Summaryp. 37
Notationp. 37
Some Definitions of Unimodality for Distributions on R[superscript n]p. 38
Logconcave Distributionsp. 46
Preservation of Unimodality Properties under Weak Limitsp. 53
The Choquet Version of Central Convex Unimodalityp. 54
Interrelationships among the Definitionsp. 57
Marginal Distributionsp. 61
Convolutionsp. 63
Some More Notions of Unimodality
Summaryp. 67
Schur Concavity and Related Concepts of Unimodalityp. 67
Generalized Unimodality Indexed by a Positive Parameterp. 72
Classes of Concave Densities and Measuresp. 84
Unimodality for Discrete Distributions
Summaryp. 101
The Structure of Unimodal Discrete Distributionsp. 101
Convolutions of Discrete Unimodal Distributions and Strong Unimodalityp. 108
Unimodality of High Convolutionsp. 112
Unimodality of Infinitely Divisible Distributions
Summaryp. 117
Infinitely Divisible Distributionsp. 117
Unimodality of Symmetric Infinitely Divisible Distributionsp. 122
The Unimodality of all Distributions in Lp. 128
The Unimodality of Multivariate Infinitely Divisible Distributionsp. 140
Unimodality and Notions of Dependence
Summaryp. 147
Preliminaries on Notions of Dependencep. 147
Monotonicity Properties of Probabilities Rectangular Regionsp. 151
Ordering of Distributions by Peakedness
Summaryp. 159
Basic Results on Peakedness Orderingp. 159
Peakedness Comparisons for the Multivariate Normal and Other Suitably Unimodal Distributionsp. 165
The Use of Unimodality and Peakedness Comparisons in Determining the Recurrence of Symmetric Random Walksp. 173
Applications of Unimodality in Statistical Inference
Summaryp. 177
Unimodality of the Likelihood Functionp. 177
Estimation of a Modep. 187
Monotonicity of the Power Functions of Certain Multivariate Testsp. 200
Shortest Confidence Intervals and Minimum Volume Confidence Regionsp. 208
Convexity in Reliability Theory
Summaryp. 215
Some Orderings for Distributions of Nonnegative Random Variablesp. 215
Univariate IFR and IFRA Classesp. 224
Multivariate IFR and IFRA Classesp. 233
Unimodality and Other Convexity Propertiesp. 243
Convex Sets in R[superscript n]p. 249
Convergence of Probability Measuresp. 252
Convex Sets of Probability Measuresp. 254
The Brunn-Minkowski Inequalityp. 259
Referencesp. 261
Author Indexp. 271
Subject Indexp. 275
Table of Contents provided by Syndetics. All Rights Reserved.

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