Marshall / Olkin / Arnold Inequalities: Theory of Majorization and Its Applications

2. Auflage 2011
ISBN: 978-0-387-68276-1
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 909 Seiten

Reihe: Springer Series in Statistics

This book's first edition has been widely cited by researchers in diverse fields. The following are excerpts from reviews. 'Inequalities: Theory of Majorization and its Applications' merits strong praise. It is innovative, coherent, well written and, most importantly, a pleasure to read. ... This work is a valuable resource!' (Mathematical Reviews). 'The authors ... present an extremely rich collection of inequalities in a remarkably coherent and unified approach. The book is a major work on inequalities, rich in content and original in organization.' (Siam Review). 'The appearance of ... Inequalities in 1979 had a great impact on the mathematical sciences. By showing how a single concept unified a staggering amount of material from widely diverse disciplines-probability, geometry, statistics, operations research, etc.-this work was a revelation to those of us who had been trying to make sense of his own corner of this material.' (Linear Algebra and its Applications). This greatly expanded new edition includes recent research on stochastic, multivariate and group majorization, Lorenz order, and applications in physics and chemistry, in economics and political science, in matrix inequalities, and in probability and statistics. The reference list has almost doubled.

Albert W. Marshall is Professor Emeritus of Statistics at the University of British Columbia. His fundamental contributions to reliability theory have had a profound effect in furthering its development. Ingram Olkin is Professor Emeritus of Statistics at Stanford University. He has made fundamental contributions in multivariate analysis, and in the development of statistical methods in meta-analysis, which have resulted in its use in many applications. Barry C. Arnold is Distinguished Professor of Statistics at the University of California, Riverside. His previous books deal with Pareto Distributions, Order Statistics, Record Values, Conditionally Specified Distributions, and the Lorenz Order.

Autoren/Hrsg.

Marshall, Albert W.

Olkin, Ingram

Arnold, Barry C.

Inhaltsverzeichnis

1;Preface and Acknowledgments
from the First Edition
;6
2;History and Preface of the
Second Edition
;10
3;Overview and Roadmap
;13
4;Contents
;15
5;Basic Notation and Terminology
;22
6;Part I: Theory of Majorization
;26
6.1;1 Introduction;27
6.1.1;A Motivation and Basic Definitions;27
6.1.2;B Majorization as a Partial Ordering;42
6.1.3;C Order-Preserving Functions;43
6.1.4;D Various Generalizations of Majorization;45
6.2;2 Doubly Stochastic Matrices;53
6.2.1;A Doubly Stochastic Matrices and Permutation Matrices
;53
6.2.2;B Characterization of Majorization Using Doubly
Stochastic Matrices;56
6.2.3;C Doubly Substochastic Matrices and Weak
Majorization;60
6.2.4;D Doubly Superstochastic Matrices and Weak
Majorization;66
6.2.5;E Orderings on D;69
6.2.6;F Proofs of Birkhoff's Theorem and Refinements;71
6.2.7;G Classes of Doubly Stochastic Matrices;76
6.2.8;H More Examples of Doubly Stochastic and Doubly Substochastic Matrices;85
6.2.9;I Properties of Doubly Stochastic Matrices;91
6.2.10;J Diagonal Equivalence of Nonnegative Matrices and Doubly Stochastic Matrices
;100
6.3;3 Schur-Convex Functions;102
6.3.1;A Characterization of Schur-Convex Functions;103
6.3.2;B Compositions Involving Schur-Convex Functions;111
6.3.3;C Some General Classes of Schur-Convex Functions;114
6.3.4;D Examples I. Sums of Convex Functions;124
6.3.5;E Examples II. Products of LogarithmicallyConcave (Convex) Functions;128
6.3.6;F Examples III. Elementary Symmetric Functions;137
6.3.7;G Symmetrization of Convex and Schur-Convex Functions: Muirhead’s Theorem
;143
6.3.8;H Schur-Convex Functions on D and TheirExtension to R n;155
6.3.9;I Miscellaneous Specific Examples;161
6.3.10;J Integral Transformations Preserving
Schur-Convexity;168
6.3.11;K Physical Interpretations of Inequalities;176
6.4;4 Equivalent Conditions for Majorization;178
6.4.1;A Characterization by Linear Transformations;178
6.4.2;B Characterization in Terms of Order-Preserving
Functions;179
6.4.3;C A Geometric Characterization;185
6.4.4;D A Characterization Involving Top Wage Earners;186
6.5;5 Preservation and Generation of Majorization;187
6.5.1;A Operations Preserving Majorization;187
6.5.2;B Generation of Majorization;207
6.5.3;C Maximal and Minimal Vectors Under Constraints;214
6.5.4;D Majorization in Integers;216
6.5.5;E Partitions;221
6.5.6;F Linear Transformations That Preserve Majorization;224
6.6;6 Rearrangements and Majorization;225
6.6.1;A Majorizations from Additions of Vectors;226
6.6.2;B Majorizations from Functions of Vectors;232
6.6.3;C Weak Majorizations from Rearrangements;235
6.6.4;D L-Superadditive Functions---Properties
and Examples;239
6.6.5;E Inequalities Without Majorization;247
6.6.6;F A Relative Arrangement Partial Order;250
7;Part II: Mathematical Applications
;262
7.1;7 Combinatorial Analysis;263
7.1.1;A Some Preliminaries on Graphs, Incidence
Matrices, and Networks;263
7.1.2;B Conjugate Sequences;265
7.1.3;C The Theorem of Gale and Ryser;269
7.1.4;D Some Applications of the Gale--Ryser Theorem;274
7.1.5;E s-Graphs and a Generalization of the Gale--Ryser Theorem
;278
7.1.6;F Tournaments;280
7.1.7;G Edge Coloring in Graphs;285
7.1.8;H Some Graph Theory Settings in Which
Majorization Plays a Role;287
7.2;8 Geometric Inequalities;288
7.2.1;A Inequalities for the Angles of a Triangle;290
7.2.2;B Inequalities for the Sides of a Triangle;295
7.2.3;C Inequalities for the Exradii and Altitudes;301
7.2.4;D Inequalities for the Sides, Exradii, and Medians;303
7.2.5;E Isoperimetric-Type Inequalities for Plane Figures;306
7.2.6;F Duality Between Triangle Inequalities and
Inequalities Involving Positive Numbers;313
7.2.7;G Inequalities for Polygons and Simplexes;314
7.3;9 Matrix Theory;316
7.3.1;A Notation and Preliminaries;317
7.3.2;B Diagonal Elements and Eigenvalues of a Hermitian Matrix;319
7.3.3;C Eigenvalues of a Hermitian Matrix and Its
Principal Submatrices;327
7.3.4;D Diagonal Elements and Singular Values;332
7.3.5;E Absolute Value of Eigenvalues and Singular Values;336
7.3.6;F Eigenvalues and Singular Values;343
7.3.7;G Eigenvalues and Singular Values of A, B,and A + B;348
7.3.8;H Eigenvalues and Singular Values of A, B, and AB;357
7.3.9;I Absolute Values of Eigenvalues and Row Sums;366
7.3.10;J Schur or Hadamard Products of Matrices;371
7.3.11;K Diagonal Elements and Eigenvalues of a Totally Positive Matrix and of an M-Matrix
;376
7.3.12;L Loewner Ordering and Majorization;379
7.3.13;M Nonnegative Matrix-Valued Functions;380
7.3.14;N Zeros of Polynomials;381
7.3.15;O Other Settings in Matrix Theory Where
Majorization Has Proved Useful;382
7.4;10 Numerical Analysis;385
7.4.1;A Unitarily Invariant Norms and Symmetric
Gauge Functions;385
7.4.2;B Matrices Closest to a Given Matrix;388
7.4.3;C Condition Numbers and Linear Equations;394
7.4.4;D Condition Numbers of Submatrices
and Augmented Matrices;398
7.4.5;E Condition Numbers and Norms;398
8;Part III: Stochastic Applications
;402
8.1;11 Stochastic Majorizations;403
8.1.1;A Introduction;403
8.1.2;B Convex Functions and Exchangeable
Random Variables;408
8.1.3;C Families of Distributions Parameterized to Preserve Symmetry and Convexity
;413
8.1.4;D Some Consequences of the Stochastic
Majorization E1(P1);417
8.1.5;E Parameterization to Preserve Schur-Convexity;419
8.1.6;F Additional Stochastic Majorizations and Properties;436
8.1.7;G Weak Stochastic Majorizations;443
8.1.8;H Additional Stochastic Weak Majorizations and Properties;451
8.1.9;I Stochastic Schur-Convexity;456
8.2;12 Probabilistic, Statistical, and Other Applications;457
8.2.1;A Sampling from a Finite Population;458
8.2.2;B Majorization Using Jensen's Inequality;472
8.2.3;C Probabilities of Realizing at Least k of n Events;473
8.2.4;D Expected Values of Ordered Random Variables;477
8.2.5;E Eigenvalues of a Random Matrix;485
8.2.6;F Special Results for Bernoulli and Geometric
Random Variables;490
8.2.7;G Weighted Sums of Symmetric Random Variables;492
8.2.8;H Stochastic Ordering from Ordered Random
Variables;497
8.2.9;I Another Stochastic Majorization Based on Stochastic Ordering;503
8.2.10;J Peakedness of Distributions of Linear Combinations;506
8.2.11;K Tail Probabilities for Linear Combinations;510
8.2.12;L Schur-Concave Distribution Functions and Survival
Functions;516
8.2.13;M Bivariate Probability Distributions with Fixed
Marginals;521
8.2.14;N Combining Random Variables;523
8.2.15;O Concentration Inequalities for Multivariate
Distributions;526
8.2.16;P Miscellaneous Cameo Appearances of Majorization;527
8.2.17;Q Some Other Settings in Which Majorization
Plays a Role;541
8.3;13 Additional Statistical Applications;543
8.3.1;A Unbiasedness of Tests and Monotonicity
of Power Functions;544
8.3.2;B Linear Combinations of Observations;551
8.3.3;C Ranking and Selection;557
8.3.4;D Majorization in Reliability Theory;565
8.3.5;E Entropy;572
8.3.6;F Measuring Inequality and Diversity;575
8.3.7;G Schur-Convex Likelihood Functions;582
8.3.8;H Probability Content of Geometric Regions
for Schur-Concave Densities;583
8.3.9;I Optimal Experimental Design;584
8.3.10;J Comparison of Experiments;586
9;Part IV: Generalizations
;591
9.1;14 Orderings Extending Majorization;592
9.1.1;A Majorization with Weights;593
9.1.2;B Majorization Relative to d;600
9.1.3;C Semigroup and Group Majorization;602
9.1.4;D Partial Orderings Induced by Convex Cones;610
9.1.5;E Orderings Derived from Function Sets;613
9.1.6;F Other Relatives of Majorization;618
9.1.7;G Majorization with Respect to a Partial Order;620
9.1.8;H Rearrangements and Majorizations for Functions;621
9.2;15 Multivariate Majorization;625
9.2.1;A Some Basic Orders;625
9.2.2;B The Order-Preserving Functions;635
9.2.3;C Majorization for Matrices of Differing Dimensions;637
9.2.4;D Additional Extensions;642
9.2.5;E Probability Inequalities;644
10;Part V: Complementary Topics
;648
10.1;16 Convex Functions and Some Classical Inequalities;649
10.1.1;A Monotone Functions;649
10.1.2;B Convex Functions;653
10.1.3;C Jensen's Inequality;666
10.1.4;D Some Additional Fundamental Inequalities;669
10.1.5;E Matrix-Monotone and Matrix-Convex Functions;682
10.1.6;F Real-Valued Functions of Matrices;696
10.2;17 Stochastic Ordering;705
10.2.1;A Some Basic Stochastic Orders;706
10.2.2;B Stochastic Orders from Convex Cones;712
10.2.3;C The Lorenz Order;724
10.2.4;D Lorenz Order: Applications and Related Results;746
10.2.5;E An Uncertainty Order;760
10.3;18 Total Positivity;769
10.3.1;A Totally Positive Functions;769
10.3.2;B Pólya Frequency Functions;774
10.3.3;C Pólya Frequency Sequences;779
10.3.4;D Total Positivity of Matrices;779
10.4;19 Matrix Factorizations, Compounds,Direct Products, and M-Matrices;781
10.4.1;A Eigenvalue Decompositions;781
10.4.2;B Singular Value Decomposition;783
10.4.3;C Square Roots and the Polar Decomposition;784
10.4.4;D A Duality Between Positive Semidefinite Hermitian Matrices and Complex Matrices
;786
10.4.5;E Simultaneous Reduction of Two Hermitian Matrices;787
10.4.6;F Compound Matrices;787
10.4.7;G Kronecker Product and Sum;792
10.4.8;H M-Matrices;794
10.5;20 Extremal Representations of Matrix Functions;795
10.5.1;A Eigenvalues of a Hermitian Matrix;795
10.5.2;B Singular Values;801
10.5.3;C Other Extremal Representations;806
11;Biographies;808
12;References;823
13;Author Index;889
14;Subject Index;903