# Gini Inequality Index: Methods and Applications

Introducing Informal Inequality Measures(IIMs) Constructed from U-statistics of Degree Three or Higher in Analyzing Economic DisparityIntroductionA Brief Review of the LiteratureA Modest Goal and the Layout of This PaperPreliminaries, Illustrations, and Economic Persuasions Behind the New IIMsSome PreliminariesIIMs of Degree 3IIMs of Degree 4Economic Persuasions and Motivations via IllustrationsIllustration 2.1: Different Income DistributionsWith Same Misleading GIllustration 2.2: Same Income Distribution with Different GIllustrations via Simulations Under Gamma andLognormal DistributionsA General Class of New IIMsSelected Properties of the New IIMsAddressing the Pigou-Dalton Transfer PropertyEmpirical Validation of Pigou-Dalton TransferMoments of IIMs With ApplicationsA Consistent Estimator of ξ Defined Via (4.1)Applications: Large-Sample Confidence Intervals for θklIllustrations With Real DataOne-Sample ProblemsTwo-Sample ProblemsConcluding ThoughtsSpecial Attention to IIMs H21 and H31Special Attention to IIM H22Last WordsAcknowledgmentsReferencesThe Decomposition of the Gini Index Between and Within Groups: A Key Factor in Gender Studies An Application in the Context of Salary Distribution in SpainIntroductionMethodology: Decomposition of the Gini Index Betweenand Within GroupsDescription of the DataResults: Evolution of Salary Concentration in Spainin the Period 2006–2014Inequality Among the Group of Workers According to TheirPersonal, Work, and Company CharacteristicsComparison of Levels of Wage Concentration Within the Groupof Women Workers and the Group of Male Workers Accordingto Their Personal, Work, and Company CharacteristicsComparison of Gender Wage Concentration Levels Accordingto Personal, Work, and Company CharacteristicsConclusionsAcknowledgmentsReferencesA Note on the Decomposition of Health Inequality by Population Subgroups in the Case of Ordinal VariablesIntroductionThe Decomposition of Health Inequalityby Population SubgroupsThe Proposal of Kobus and Miloś (2012)The Gini-Related Index of Lv et al. (2015)The Properties of the Index Introduced by Lv et al. (2015)Decomposing by Population Subgroups the Gini-Related IndexProposed by Lv et al. (2015)An Empirical IllustrationReferencesThe Gini Index Decomposition and Overlapping Between Population SubgroupsIntroductionOverlappingThe Measurement of OverlappingThe Probability of TransvariationThe Intensity of TransvariationAn Illustrative ExampleThe Gini Index DecompositionInequality WithinInequality Between and Overlapping ComponentMean-Based EvaluationsDistribution-Based EvaluationsThe Comparison of DecompositionsAn Illustrative ExampleInequality Decomposition, Overlapping, and Political Economy: The Analysis of Gender GapAn Illustrative ExampleA Case Study: The Italian Personal Income by GenderConclusionsReferencesGini's Mean Difference-Based Minimum Risk Point Estimator of MeanIntroductionProblem FormulationContributionPurely Sequential ProcedurePilot Sample Size ComputationCharacteristicsSimulation StudyConclusionReferencesThe Gini Concentration Index for the Studyof SurvivalIntroductionEstimation of the Gini Concentration Index from Incomplete DataSome Types of Incomplete Survival (or Income) DataParametric and Nonparametric EstimationThe Restricted Gini Index and TestEstimation with Dependent CensoringThe Gini Concentration Index for the Study of Survival in DemographyNonhuman PopulationsDecomposition, Forecasting, and Interpretation of InequalityA Family of Survival Models for Longevity and ConcentrationFinal CommentReferencesAn Axiomatic Analysis of Generalized Gini Air Quality IndicesIntroductionSingle-Pollutant Air Quality Indices: An Illustrative DiscussionAxioms for a Composite Air Quality IndexComposite Air Quality Indices: A Brief Illuminating DiscussionThe Characterization TheoremsConclusionsAcknowledgmentsReferencesSequential Confidence Set and Point Estimation of the Population Gini Index by Controlling Accuracies Relative to the Population MeanIntroductionRevised Loss FunctionsAn Overview and the Layout of the PaperRelative-Accuracy Confidence Set EstimationPurely Sequential Sampling MethodologyAsymptotic First-Order PropertiesSimulation StudiesConfidence Set EstimationPoint EstimationAppendix with Selected TechnicalitiesProof of Theorem 8.3Proof of Theorem 8.4ReferencesA Test on Correlation Based on Gini's Mean DifferenceIntroductionTesting on CorrelationAnalysis of the GMD for Correlated VariablesTests Based on the GMDAnalysis of the Power Function of the Test Based on Tn(1)Comparison of Several Tests Based on the GMDApplication in Statistical Process ControlConclusionsReferencesSegregation Measures for Different Forms of Categorical Data: Reinterpretation and ProposalIntroductionSegregation Measure for Nominal Categorical DataThe Set of Axiomatic Properties Required in the Analysis of SegregationMeasures Defined from the Concept of AssociationMeasure of Segregation Constructed from Unequal RepresentationSegregation Measures for Ordinal Categorical DataAn Axiomatic Characterization of the Segregation MeasuresMeasure Defined from the Concept of AssociationMeasure Defined from the Concept of Unequal RepresentationConclusionAppendix IAppendix IIAppendix IIIAppendix IVReferencesExploring Fixed-Accuracy Estimationfor Population Gini Inequality Index Under Big Data: A Passage to Practical Distribution-Free StrategiesIntroductionRecent Developments in Sequential Estimation StrategiesFixed-Width Confidence Interval (FWCI) StrategyMinimum Risk Point Estimation (MRPE) StrategyBig Data EraA Broader OverviewThe Layout of the ChapterNew FWCI and MRPE Formulations Under Big DataThe Foundation and StructureThe FWCI Problem: Determination of the Optimal Number rThe MRPE Problem: Determination of the Optimal Number rA Suggested Guide for Choices of kEstimation of the Asymptotic VarianceSequential Estimation Strategy for the FWCI ProblemAsymptotic First-Order ResultsAsymptotic Normality of Stopping TimeHeuristics on Asymptotic Second-Order Results: A Practical GuideSequential Estimation Strategy for the MRPE ProblemAsymptotic First-Order ResultsAsymptotic Second-Order Results: A Brief OutlineConcluding Thoughts: Flexibility of the Proposed Approachin Big Data ScienceAcknowledgmentsReferences