Gini Inequality Index Methods and Applications

Cover
Title Pages
Half Title
Copyright Page
Dedication Page
Contents
Foreword: Giovanni Maria Gior
Foreword: Shelemyahu Zacks
Foreword: K.V. Mardia
Preface
Contributors
1 Introducing Informal Inequality Measures(IIMs) Constructed from U-statistics of Degree Three or Higher in Analyzing Economic Disparity
	1.1 Introduction
		1.1.1 A Brief Review of the Literature
		1.1.2 A Modest Goal and the Layout of This Paper
	1.2 Preliminaries, Illustrations, and Economic Persuasions Behind the New IIMs
		1.2.1 Some Preliminaries
			1.2.1.1 IIMs of Degree 3
			1.2.1.2 IIMs of Degree 4
		1.2.2 Economic Persuasions and Motivations via Illustrations
			1.2.2.1 Illustration 2.1: Different Income DistributionsWith Same Misleading G
			1.2.2.2 Illustration 2.2: Same Income Distribution with Different G
		1.2.3 Illustrations via Simulations Under Gamma andLognormal Distributions
	1.3 A General Class of New IIMs
		1.3.1 Selected Properties of the New IIMs
		1.3.2 Addressing the Pigou-Dalton Transfer Property
			1.3.2.1 Empirical Validation of Pigou-Dalton Transfer
	1.4 Moments of IIMs With Applications
		1.4.1 A Consistent Estimator of ξ Defined Via (4.1)
		1.4.2 Applications: Large-Sample Confidence Intervals for  θkl
	1.5 Illustrations With Real Data
		1.5.1 One-Sample Problems
		1.5.2 Two-Sample Problems
	1.6 Concluding Thoughts
		1.6.1 Special Attention to IIMs H21 and H31
		1.6.2 Special Attention to IIM H22
		1.6.3 Last Words
	Acknowledgments
	References
2 The Decomposition of the Gini Index Between and Within Groups: A Key Factor in Gender Studies An Application in the Context of Salary Distribution in Spain
	2.1 Introduction
	2.2 Methodology: Decomposition of the Gini Index Betweenand Within Groups
	2.3 Description of the Data
	2.4 Results: Evolution of Salary Concentration in Spainin the Period 2006–2014
		2.4.1 Inequality Among the Group of Workers According to TheirPersonal, Work, and Company Characteristics
		2.4.2 Comparison of Levels of Wage Concentration Within the Groupof Women Workers and the Group of Male Workers Accordingto Their Personal, Work, and Company Characteristics
		2.4.3 Comparison of Gender Wage Concentration Levels Accordingto Personal, Work, and Company Characteristics
	2.5 Conclusions
	Acknowledgments
	References
3 A Note on the Decomposition of Health Inequality by Population Subgroups in the Case of Ordinal Variables
	3.1 Introduction
	3.2 The Decomposition of Health Inequalityby Population Subgroups
		3.2.1 The Proposal of Kobus and Miloś (2012)
		3.2.2 The Gini-Related Index of Lv et al. (2015)
			3.2.2.1 The Properties of the Index Introduced by Lv et al. (2015)
			3.2.2.2 Decomposing by Population Subgroups the Gini-Related IndexProposed by Lv et al. (2015)
	3.3 An Empirical Illustration
	References
4 The Gini Index Decomposition and Overlapping Between Population Subgroups
	4.1 Introduction
	4.2 Overlapping
		4.2.1 The Measurement of Overlapping
			4.2.1.1 The Probability of Transvariation
			4.2.1.2 The Intensity of Transvariation
		4.2.2 An Illustrative Example
	4.3 The Gini Index Decomposition
		4.3.1 Inequality Within
		4.3.2 Inequality Between and Overlapping Component
			4.3.2.1 Mean-Based Evaluations
			4.3.2.2 Distribution-Based Evaluations
		4.3.3 The Comparison of Decompositions
		4.3.4 An Illustrative Example
	4.4 Inequality Decomposition, Overlapping, and Political Economy: The Analysis of Gender Gap
		4.4.1 An Illustrative Example
		4.4.2 A Case Study: The Italian Personal Income by Gender
	4.5 Conclusions
	References
5 Gini's Mean Difference-Based Minimum Risk Point Estimator of Mean
	5.1 Introduction
		5.1.1 Problem Formulation
		5.1.2 Contribution
	5.2 Purely Sequential Procedure
		5.2.1 Pilot Sample Size Computation
	5.3 Characteristics
	5.4 Simulation Study
	5.5 Conclusion
	References
6 The Gini Concentration Index for the Studyof Survival
	6.1 Introduction
	6.2 Estimation of the Gini Concentration Index from Incomplete Data
		6.2.1 Some Types of Incomplete Survival (or Income) Data
		6.2.2 Parametric and Nonparametric Estimation
			6.2.2.1 The Restricted Gini Index and Test
		6.2.3 Estimation with Dependent Censoring
	6.3 The Gini Concentration Index for the Study of Survival in Demography
		6.3.1 Nonhuman Populations
		6.3.2 Decomposition, Forecasting, and Interpretation of Inequality
	6.4 A Family of Survival Models for Longevity and Concentration
	6.5 Final Comment
	References
7 An Axiomatic Analysis of Generalized Gini Air Quality Indices
	7.1 Introduction
	7.2 Single-Pollutant Air Quality Indices: An Illustrative Discussion
	7.3 Axioms for a Composite Air Quality Index
	7.4 Composite Air Quality Indices: A Brief Illuminating Discussion
	7.5 The Characterization Theorems
	7.6 Conclusions
	Acknowledgments
	References
8 Sequential Confidence Set and Point Estimation of the Population Gini Index by Controlling Accuracies Relative to the Population Mean
	8.1 Introduction
		8.1.1 Revised Loss Functions
		8.1.2 An Overview and the Layout of the Paper
	8.2 Relative-Accuracy Confidence Set Estimation
		8.2.1 Purely Sequential Sampling Methodology
		8.2.2 Asymptotic First-Order Properties
	8.3 Minimum Relative Risk Point Estimation (MRRPE)
	8.4 Simulation Studies
		8.4.1 Confidence Set Estimation
		8.4.2 Point Estimation
	8.5 Appendix with Selected Technicalities
		8.5.1 Proof of Theorem 8.3
		8.5.2 Proof of Theorem 8.4
	Acknowledgments
	References
9 A Test on Correlation Based on Gini's Mean Difference
	9.1 Introduction
	9.2 Testing on Correlation
		9.2.1 Analysis of the GMD for Correlated Variables
		9.2.2 Tests Based on the GMD
		9.2.3 Analysis of the Power Function of the Test Based on Tn(1)
		9.2.4 Comparison of Several Tests Based on the GMD
	9.3 Application in Statistical Process Control
	9.4 Conclusions
	References
10 Segregation Measures for Different Forms of Categorical Data: Reinterpretation and Proposal
	10.1 Introduction
	10.2 Segregation Measure for Nominal Categorical Data
		10.2.1 Basic Notations and Definitions
		10.2.2 The Set of Axiomatic Properties Required in the Analysis of Segregation
		10.2.3 Measures Defined from the Concept of Association
		10.2.4 Measure of Segregation Constructed from Unequal Representation
	10.3 Segregation Measures for Ordinal Categorical Data
		10.3.1 Basic Notations and Definition
		10.3.2 An Axiomatic Characterization of the Segregation Measures
		10.3.3 Measure Defined from the Concept of Association
		10.3.4 Measure Defined from the Concept of Unequal Representation
	10.4 Conclusion
	Appendix I
		Proof of the Proposition 1
	Appendix II
		Proof of the Proposition 2
	Appendix III
		Proof of the Proposition 3
	Appendix IV
		Proof of the Proposition 4
	References
11 Exploring Fixed-Accuracy Estimationfor Population Gini Inequality Index Under Big Data: A Passage to Practical Distribution-Free Strategies
	11.1 Introduction
		11.1.1 Recent Developments in Sequential Estimation Strategies
			11.1.1.1 Fixed-Width Confidence Interval (FWCI) Strategy
			11.1.1.2 Minimum Risk Point Estimation (MRPE) Strategy
		11.1.2 Big Data Era
		11.1.3 A Broader Overview
		11.1.4 The Layout of the Chapter
	11.2 New FWCI and MRPE Formulations Under Big Data
		11.2.1 The Foundation and Structure
		11.2.2 The FWCI Problem: Determination of the Optimal Number r
		11.2.3 The MRPE Problem: Determination of the Optimal Number r
		11.2.4 A Suggested Guide for Choices of k
		11.2.5 Estimation of the Asymptotic Variance
	11.3 Sequential Estimation Strategy for the FWCI Problem
		11.3.1 Asymptotic First-Order Results
		11.3.2 Asymptotic Normality of Stopping Time
		11.3.3 Heuristics on Asymptotic Second-Order Results: A Practical Guide
	11.4 Sequential Estimation Strategy for the MRPE Problem
		11.4.1 Asymptotic First-Order Results
		11.4.2 Asymptotic Second-Order Results: A Brief Outline
	11.5 Concluding Thoughts: Flexibility of the Proposed Approachin Big Data Science
	Acknowledgments
	References
Index