Majorization and the Lorenz Order with Applications in Applied Mathematics and Economics
by Barry C. Arnold (Author), José María Sarabia (Author)

About the author
Barry C. Arnold, PhD, is Distinguished Professor in the Statistics Department at the University of California, Riverside. He is a Fellow of the American Statistical Society, the American Association for the Advancement of Science, and the Institute of Mathematical Statistics, and is an elected member of the International Statistical Institute. He is the author of more than two hundred publications and eight books.
José María Sarabia, PhD, is Statistics Professor in the Department of Economics at the University of Cantabria, Spain. He is author of more than one hundred publications and ten books and is an associate editor of several journals, including Journal of Banking and Finance, TEST, and Journal of Statistical Distributions and Applications.

About this book
This book was written to serve as a graduate-level textbook for special topics classes in mathematics, statistics, and economics, to introduce these topics to other researchers, and for use in short courses. It is an introduction to the theory of majorization and related notions, and contains detailed material on economic applications of majorization and the Lorenz order, investigating the theoretical aspects of these two interrelated orderings.
Revising and expanding on an earlier monograph, Majorization and the Lorenz Order: A Brief Introduction, the authors provide a straightforward development and explanation of majorization concepts, addressing historical development of the topics, and providing up-to-date coverage of families of Lorenz curves. The exposition of multivariate Lorenz orderings sets it apart from existing treatments of these topics.
Mathematicians, theoretical statisticians, e conomists, and other social scientists who already recognize the utility of the Lorenz order in income inequality contexts and arenas will find the book useful for its sound development of relevant concepts rigorously linked to both the majorization literature and the even more extensive body of research on economic applications.

Table of contents
1 Introduction
    1. 1 Early Work About Majorization
    1. 2 The Definition of Majorization
2 Majorization in Rn
    2. 1 Basic Result
    2. 2 Schur Convex Functions and Majorization
    2. 3 Exercises
3 The Lorenz Order in the Space of Distribution Functions
    3. 1 The Lorenz Curve
    3. 2 The Lorenz Order
    3. 3 Exercises
4 Transformations and Their Effects
    4. 1 Deterministic Transformations
    4. 2 Stochastic Transformations
    4. 3 Exercises
5 Inequality Measures
    5. 1 Introduction
    5. 2 Common Measures of Inequality
        5. 2. 1 Seven Basic Inequality Measures
        5. 2. 2 Inequality Measures Based on the Concept of Entropy
    5. 3 Inequality Measures Derived from the Lorenz Curve
        5. 3. 1 The Gini Index
        5. 3. 2 Generalizations of the Gini Index
        5. 3. 3 Decomposition of the Gini and Yitzhaki Indices
        5. 3. 4 Inequality Indices Related to Lorenz Curve Moments
        5. 3. 5 The Pietra Index
        5. 3. 6 The Palma Index and Income Share Ratios Inequality Indices
        5. 3. 7 The Amato Index
        5. 3. 8 The Elteto and Frigyes Inequality Measures
    5. 4 The Atkinson and the Generalized Entropy Indices
        5. 4. 1 The Atkinson Indices
        5. 4. 2 The Generalized Entropy Indices and the Theil Indices
        5. 4. 3 Decomposability of Certain Indices
    5. 5 Estimation with Partial Information
        5. 5. 1 Bounds on the Gini Index
        5. 5. 2 Parameter Identification Using the Mean and the Gini Index
    5. 6 Moment Distributions
    5. 7 Relations Between Inequality Measures
    5. 8 Sample Versions of Analytic Measures of Inequality
        5. 8. 1 Absolute and Relative Mean Deviation and the Sample Pietra Index
        5. 8. 2 The Sample Amato and Bonferroni Indices
        5. 8. 3 The Sample Standard Deviation and Coefficient of Variation
        5. 8. 4 Gini’s Mean Difference
        5. 8. 5 The Sample Gini Index
        5. 8. 6 Sample Lorenz Curve
        5. 8. 7 Bias of the Sample Lorenz Curve and Gini Index
        5. 8. 8 Asymptotic Distribution of Lorenz Ordinates and Income Shares
        5. 8. 9 The Elteto and Frigyes Indices
        5. 8. 10 Further Classical Sample Measures of Inequality
        5. 8. 11 The Sample Atkinson and Generalized Entropy Indices
        5. 8. 12 The Kolm Inequality Indices
        5. 8. 13 Additional Sample Inequality Indices
    5. 9 A New Class of Inequality Measures
    5. 10 Exercises
6 Families of Lorenz Curves
    6. 1 Basic Results
        6. 1. 1 A Characterization of the Lorenz Curve
        6. 1. 2 Lorenz Curves of Some Common Distributions
        6. 1. 3 Translated and Truncated Lorenz Curves
        6. 1. 4 The Modality of the Income Density Function
    6. 2 The Alchemy of Lorenz Curves
    6. 3 Parametric Families of Lorenz Curves
        6. 3. 1 Some Hierarchical Families
        6. 3. 2 General Quadratic Lorenz Curves
        6. 3. 3 Other Parametric Families
    6. 4 Some Alternative Inequality Curves
        6. 4. 1 Generalized and Absolute Lorenz Curves
        6. 4. 2 Leimkuhler, Bonferroni and Zenga Curves
        6. 4. 3 Inequality Curves for the Lower and Middle Income Groups
        6. 4. 4 Reliability Curves
        6. 4. 5 Relative Deprivation
    6. 5 Exercises
7 Multivariate Majorization and Multivariate Lorenz Ordering
    7. 1 Multivariate Majorization
    7. 2 Multivariate Lorenz Orderings
    7. 3 Explicit Expressions for the Arnold Lorenz Surface
        7. 3. 1 The Bivariate Sarmanov–Lee Lorenz Surface
    7.4 Summary Measures of m -Dimensional Inequality  
        7. 4. 1 Bivariate Gini Index for the Arnold Lorenz Surface
    7. 5 Alternative Multivariate Inequality Indices
        7. 5. 1 Multivariate Shannon and Rényi Entropies
        7. 5. 2 Multivariate Generalized Entropy and Theil Indices
    7. 6 Exercises
8 Stochastic Majorization
    8. 1 Definition and Main Results
    8. 2 Exercises
9 Some Related Orderings
    9. 1 Star-Ordering
    9. 2 Stochastic Dominance
    9. 3 Exercises
10 Inequality Analysis in Families of Income Distributions
    10. 1 Introduction
    10. 2 The McDonald Family: Definitions and Basic Properties
        10. 2. 1 Lorenz Curves and Gini Indices
        10. 2. 2 Other Inequality Measures
    10. 3 The Generalized Pareto Distributions
        10. 3. 1 Lorenz Curves and Gini Indices
        10. 3. 2 Inequality Measures
    10. 4 Stochastic Orderings Within the McDonald Family
        10. 4. 1 Introduction and the Orderings to be Used
        10. 4. 2 Comparisons for Two Distributions in the Same Subfamily of McDonald Distributions
        10. 4. 3 Comparisons for Two Distributions in Different McDonald Subfamilies
    10. 5 Exercises
11 Some Applications
    11. 1 A Geometric Inequality of Cesaro
    11. 2 Matrices with Prescribed Characteristic Roots
    11. 3 Variability of Sample Medians and Means
    11. 4 Reliability
    11. 5 Genetic Selection
    11. 6 Large Interactions
    11. 7 Unbiased Tests
    11.8 Summation Modulo m  
    11. 9 Forecasting
    11. 10 Ecological Diversity
    11. 11 Covering a Circle
    11. 12 Waiting for a Pattern
    11. 13 Paired Comparisons
    11. 14 Phase Type Distributions
    11. 15 Gaussian Correlation
12 More Applications
    12. 1 Catchability
    12. 2 Server Assignment Policies in Queueing Networks
    12. 3 Disease Transmission
    12. 4 Apportionment in Proportional Representation
    12. 5 Connected Components in a Random Graph
    12. 6 A Stochastic Relation Between the Sum and the Maximum of Two Random Variables
    12. 7 Segregation
    12. 8 Lorenz Order with Common Finite Support
    12. 9 The Scarsini Dependence Order
        12.9.1 Extension to k Dimensions  

Series: Statistics for Social and Behavioral Sciences
Length: 272 pages
Publisher: Springer; 1st ed. 2018 edition (September 11, 2018)
Language: English
ISBN-10: 3319937723
ISBN-13: 978-3319937724

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