Inequalities：theory of majorization and its applications.pdf

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Cover

Springer Series in Statistics

Inequalities: Theory of Majorization and Its Applications (Second edition)

9780387400877

Preface and Acknowledgments from the First Edition

History and Preface of the Second Edition

Overview and Roadmap

Contents

Basic Notation and Terminology

Part I: Theory of Majorization

1 Introduction

A Motivation and Basic Definitions

B Majorization as a Partial Ordering

C Order-Preserving Functions

D Various Generalizations of Majorization

2 Doubly Stochastic Matrices

A Doubly Stochastic Matrices and Permutation Matrices

B Characterization of Majorization Using Doubly Stochastic Matrices

C Doubly Substochastic Matrices and Weak Majorization

D Doubly Superstochastic Matrices and Weak Majorization

E Orderings on \mathscr{D}

F Proofs of Birkhoff's Theorem and Refinements

G Classes of Doubly Stochastic Matrices

H More Examples of Doubly Stochastic and Doubly Substochastic Matrices

I Properties of Doubly Stochastic Matrices

J Diagonal Equivalence of Nonnegative Matrices and Doubly Stochastic Matrices

3 Schur-Convex Functions

A Characterization of Schur-Convex Functions

B Compositions Involving Schur-Convex Functions

C Some General Classes of Schur-Convex Functions

D Examples I. Sums of Convex Functions

E Examples II. Products of Logarithmically Concave (Convex) Functions

F Examples III. Elementary Symmetric Functions

Symmetrization of Convex and Schur-Convex Functions: Muirhead’s Theorem

H Schur-Convex Functions on \mathscr{D} and Their Extension to \mathscr{R}^n

I Miscellaneous Specific Examples

J Integral Transformations Preserving Schur-Convexity

K Physical Interpretations of Inequalities

4 Equivalent Conditions for Majorization

A Characterization by Linear Transformations

B Characterization in Terms of Order-Preserving Functions

C A Geometric Characterization

D A Characterization Involving Top Wage Earners

5 Preservation and Generation of Majorization

A Operations Preserving Majorization

B Generation of Majorization

C Maximal and Minimal Vectors Under Constraints

D Majorization in Integers

E Partitions

F Linear Transformations That Preserve Majorization

6 Rearrangements and Majorization

A Majorizations from Additions of Vectors

B Majorizations from Functions of Vectors

C Weak Majorizations from Rearrangements

D L-Superadditive Functions---Properties and Examples

E Inequalities Without Majorization

F A Relative Arrangement Partial Order

Part II: Mathematical Applications

7 Combinatorial Analysis

A Some Preliminaries on Graphs, Incidence Matrices, and Networks

B Conjugate Sequences

C The Theorem of Gale and Ryser

D Some Applications of the Gale–Ryser Theorem

E s-Graphs and a Generalization of the Gale–Ryser Theorem

F Tournaments

G Edge Coloring in Graphs

H Some Graph Theory Settings in Which Majorization Plays a Role

8 Geometric Inequalities

A Inequalities for the Angles of a Triangle

B Inequalities for the Sides of a Triangle

C Inequalities for the Exradii and Altitudes

D Inequalities for the Sides, Exradii, and Medians

E Isoperimetric-Type Inequalities for Plane Figures

F Duality Between Triangle Inequalities and Inequalities Involving Positive Numbers

G Inequalities for Polygons and Simplexes

9 Matrix Theory

A Notation and Preliminaries

B Diagonal Elements and Eigenvalues of a Hermitian Matrix

C Eigenvalues of a Hermitian Matrix and Its Principal Submatrices

D Diagonal Elements and Singular Values

E Absolute Value of Eigenvalues and Singular Values

F Eigenvalues and Singular Values

G Eigenvalues and Singular Values of A, B,and A + B

H Eigenvalues and Singular Values of A, B, and AB

I Absolute Values of Eigenvalues and Row Sums, and Variations of Hadamard’s Inequality

J Schur or Hadamard Products of Matrices

K Diagonal Elements and Eigenvalues of a Totally Positive Matrix and of an M-Matrix

L Loewner Ordering and Majorization

M Nonnegative Matrix-Valued Functions

N Zeros of Polynomials

O Other Settings in Matrix Theory Where Majorization Has Proved Useful

10 Numerical Analysis

A Unitarily Invariant Norms and Symmetric Gauge Functions

B Matrices Closest to a Given Matrix

C Condition Numbers and Linear Equations

D Condition Numbers of Submatrices and Augmented Matrices

E Condition Numbers and Norms

Part III: Stochastic Applications

11 Stochastic Majorizations

A Introduction

B Convex Functions and Exchangeable Random Variables

C Families of Distributions Parameterized to Preserve Symmetry and Convexity

D Some Consequences of the Stochastic Majorization E_1(P_1)

E Parameterization to Preserve Schur-Convexity

F Additional Stochastic Majorizations and Properties

G Weak Stochastic Majorizations

H Additional Stochastic Weak Majorizations and Properties

I Stochastic Schur-Convexity

12 Probabilistic, Statistical, and Other Applications

A Sampling from a Finite Population

B Majorization Using Jensen's Inequality

C Probabilities of Realizing at Least k of n Events

D Expected Values of Ordered Random Variables

E Eigenvalues of a Random Matrix

F Special Results for Bernoulli and Geometric Random Variables

G Weighted Sums of Symmetric Random Variables

H Stochastic Ordering from Ordered Random Variables

I Another Stochastic Majorization Based on Stochastic Ordering

J Peakedness of Distributions of Linear Combinations

K Tail Probabilities for Linear Combinations

L Schur-Concave Distribution Functions and Survival Functions

M Bivariate Probability Distributions with Fixed Marginals

N Combining Random Variables

O Concentration Inequalities for Multivariate Distributions

P Miscellaneous Cameo Appearances of Majorization

Q Some Other Settings in Which Majorization Plays a Role

13 Additional Statistical Applications

A Unbiasedness of Tests and Monotonicity of Power Functions

B Linear Combinations of Observations

C Ranking and Selection

D Majorization in Reliability Theory

E Entropy

F Measuring Inequality and Diversity

G Schur-Convex Likelihood Functions

H Probability Content of Geometric Regions for Schur-Concave Densities

I Optimal Experimental Design

J Comparison of Experiments

Part IV: Generalizations

14 Orderings Extending Majorization

A Majorization with Weights

B Majorization Relative to d

C Semigroup and Group Majorization

D Partial Orderings Induced by Convex Cones

E Orderings Derived from Function Sets

F Other Relatives of Majorization

G Majorization with Respect to a Partial Order

H Rearrangements and Majorizations for Functions

15 Multivariate Majorization

A Some Basic Orders

B The Order-Preserving Functions

C Majorization for Matrices of Differing Dimensions

D Additional Extensions

E Probability Inequalities

Part V: Complementary Topics

16 Convex Functions and Some Classical Inequalities

A Monotone Functions

B Convex Functions

C Jensen's Inequality

D Some Additional Fundamental Inequalities

E Matrix-Monotone and Matrix-Convex Functions

F Real-Valued Functions of Matrices

17 Stochastic Ordering

A Some Basic Stochastic Orders

B Stochastic Orders from Convex Cones

C The Lorenz Order

D Lorenz Order: Applications and Related Results

E An Uncertainty Order

18 Total Positivity

A Totally Positive Functions

B Pólya Frequency Functions

C Pólya Frequency Sequences

D Total Positivity of Matrices

19 Matrix Factorizations, Compounds, Direct Products, and M-Matrices

A Eigenvalue Decompositions

B Singular Value Decomposition

C Square Roots and the Polar Decomposition

D A Duality Between Positive Semidefinite Hermitian Matrices and Complex Matrices

E Simultaneous Reduction of Two Hermitian Matrices

F Compound Matrices

G Kronecker Product and Sum

H M-Matrices

20 Extremal Representations of Matrix Functions

A Eigenvalues of a Hermitian Matrix

B Singular Values

C Other Extremal Representations

Biographies

References

Author Index

Subject Index

Springer Series in Statistics Advisors: P. Bickel, P. Diggle, S. Feinberg, U. Gather, I. Olkin, S. Zeger For other titles published in this series, go to http://www.springer.com/series/692

Albert W. Marshall • Ingram Olkin Barry C. Arnold Inequalities: Theory of Majorization and Its Applications Second Edition ABC

Barry C. Arnold Department of Statistics University of California Riverside, CA 92521 USA barry.arnold@ucr.edu Albert W. Marshall Department of Statistics University of British Columbia Vancouver, BC V6T 1Z2 Canada and mailing address 2781 W. Shore Drive Lummi Island, WA 98262 almarshall@earthlink.net Ingram Olkin Department of Statistics Stanford University Stanford, CA 94305 USA iolkin@stat.stanford.edu ISSN 0172-7397 ISBN 978-0-387-40087-7 DOI 10.1007/978-0-387-68276-1 Springer New York Dordrecht Heidelberg London e-ISBN 978-0-387-68276-1 Library of Congress Control Number: 2010931704 c Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identiﬁed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To our long-suﬀering wives for their patience with this project: Sheila (AWM), Anita (IO), Carole (BCA) To the memory of Z.W. (Bill) Birnbaum and Edwin Hewitt who initiated my interest in inequalities (AWM) To my students and colleagues who have energized, enriched and enlivened my life (IO) To the memory of Peggy Franklin (BCA)

Preface and Acknowledgments from the First Edition Preface Although they play a fundamental role in nearly all branches of math- ematics, inequalities are usually obtained by ad hoc methods rather than as consequences of some underlying “theory of inequalities.” For certain kinds of inequalities, the notion of majorization leads to such a theory that is sometimes extremely useful and powerful for deriv- ing inequalities. Moreover, the derivation of an inequality by methods of majorization is often very helpful both for providing a deeper understanding and for suggesting natural generalizations. As the 1960s progressed, we became more and more aware of these facts. Our awareness was reinforced by a series of seminars we gave while visiting the University of Cambridge in 1967–1968. Because the ideas associated with majorization deserve to be better known, we decided by 1970 to write a little monograph on the subject—one that might have as many as 100 pages—and that was the genesis of this book. The idea of majorization is a special case of several more general no- tions, but these generalizations are mentioned in this book only for the perspective they provide. We have limited ourselves to various aspects of majorization partly because we want to emphasize its importance and partly because its simplicity appeals to us. However, to make the vii

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