Statistics for Social and Behavioral Sciences Barry C. Arnold · José María Sarabia Majorization and the Lorenz Order with Applications in Applied Mathematics and Economics Statistics for Social and Behavioral Sciences SeriesEditor StephenE.Fienberg(inmemorium) CarnegieMellonUniversity,Pittsburgh,Pennsylvania,USA Moreinformationaboutthisseriesathttp://www.springernature.com/series/3463 Barry C. Arnold • José María Sarabia Majorization and the Lorenz Order with Applications in Applied Mathematics and Economics 123 BarryC.Arnold JoséMaríaSarabia DepartmentofStatistics DepartmentofEconomics UniversityofCalifornia UniversityofCantabria Riverside,CA,USA Santander,Spain ISSN2199-7357 ISSN2199-7365 (electronic) StatisticsforSocialandBehavioralSciences ISBN978-3-319-93772-4 ISBN978-3-319-93773-1 (eBook) https://doi.org/10.1007/978-3-319-93773-1 LibraryofCongressControlNumber:2018944630 ©SpringerInternationalPublishingAG,partofSpringerNature2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland ToDarrel,LisaandKaelyn(BCA) ToCori,JoséMaríaandBelén(JMS) Preface to the Second Edition In the 31 years since the first edition was published, there has been considerable growth in the volume of research dealing with inequality and the Lorenz order. This is especially true, quite naturally in the Econometrics literature. Robin Hood retains a significant presence, and majorization still provides strong motivation for consideration of Lorenz curves in discussions of inequality. There has been significant growth in the number of summary measures of inequality that are now considered,andmuchmoreattentionisnowpaidtodiscussionofflexibleparametric familiesofLorenzcurves.Inthepresentedition,suchissuesareaddressedinsome depth in Chaps.5, 6 and 10. In the current Chap.7 will be found a more detailed discussionofmultivariateLorenzorderingthanwasincludedinChap.5ofthefirst edition. The important concept of a Lorenz zonoid is prominent in the discussion. Alternative Lorenz surfaces are also considered. The first edition ended with a chapterwhichincludedapotpourriofsituationsinwhich,sometimessurprisingly, majorizationand/orLorenzorderingplayedan,oftenimportant,roleintheanalysis. Overtheyears,thenumberofsuchexampleshasnaturallyincreased.Therewereten examplesofsuchapplicationsinthefinalchapterofthefirstedition.Therearenow 24,spreadovertwochapters.Theseexamplesreinforcethebeliefthatmajorization and Lorenz ordering continue to find new areas of application often only faintly related to income inequality but where variability comparisons are of importance. Going back to our hero, Robin Hood, pictured on page 7, wherever inequality (or variability)istobefound,RobinHoodwillappear. We are grateful to many colleagues for helpful discussion of material in this book, and we are grateful to the editorial staff of Springer for their patience and encouragementwhilethebookwasdeveloping. Riverside,CA,USA BarryC.Arnold Santander,Cantabria,Spain JoséMaríaSarabia March21,2018 vii Preface to the First Edition My interest in majorization was first spurred by Ingram Olkin’s proclivity for finding Schur convex functions lurking in the problem section of every issue of theAmericanMathematicalMonthly.Latermyinterestinincomeinequalityledme again to try and “really” understand Hardy, Littlewood and Polya’s contributions to the majorization literature. I have found the income distribution context to be quite convenient for discussion of inequality orderings. The present set of notes isdesignedforaonequartercourseintroducingmajorizationandtheLorenzorder. TheinequalityprinciplesofDalton,especiallythetransferorRobinHoodprinciple, aregivenappropriateprominence. Initial versions of this material were used in graduate statistics classes taught at the Colegio de Postgraduados, Montecillos, Mexico, and the University of California,Riverside.Iamgratefultostudentsintheseclassesfortheirconstructive critical commentaries. My wife Carole made noble efforts to harness my free- form writing and punctuation. Occasionally I was unmoved by her requests for clarification. Time will probably prove her right in these instances also. Peggy Franklin did an outstanding job of typing the manuscript, and patiently endured requestsforinnumerablemodifications. Riverside,CA,USA BarryC.Arnold July1986 ix Contents 1 Introduction................................................................. 1 1.1 EarlyWorkAboutMajorization.................................... 1 1.2 TheDefinitionofMajorization..................................... 5 2 MajorizationinR+n ......................................................... 9 2.1 BasicResult......................................................... 9 2.2 SchurConvexFunctionsandMajorization........................ 13 2.3 Exercises ............................................................ 21 3 TheLorenzOrderintheSpaceofDistributionFunctions............. 23 3.1 TheLorenzCurve................................................... 24 3.2 TheLorenzOrder................................................... 27 3.3 Exercises ............................................................ 32 4 TransformationsandTheirEffects....................................... 35 4.1 DeterministicTransformations..................................... 35 4.2 StochasticTransformations......................................... 38 4.3 Exercises ............................................................ 41 5 InequalityMeasures........................................................ 45 5.1 Introduction ......................................................... 45 5.2 CommonMeasuresofInequality .................................. 46 5.2.1 SevenBasicInequalityMeasures......................... 46 5.2.2 InequalityMeasuresBasedontheConceptofEntropy . 49 5.3 InequalityMeasuresDerivedfromtheLorenzCurve............. 51 5.3.1 TheGiniIndex ............................................ 51 5.3.2 GeneralizationsoftheGiniIndex ........................ 54 5.3.3 DecompositionoftheGiniandYitzhakiIndices ........ 56 5.3.4 InequalityIndicesRelatedtoLorenzCurveMoments .. 61 5.3.5 ThePietraIndex........................................... 63 5.3.6 The Palma Index and Income Share Ratios InequalityIndices ......................................... 66 xi xii Contents 5.3.7 TheAmatoIndex.......................................... 67 5.3.8 TheEltetoandFrigyesInequalityMeasures............. 68 5.4 TheAtkinsonandtheGeneralizedEntropyIndices............... 69 5.4.1 TheAtkinsonIndices ..................................... 70 5.4.2 TheGeneralizedEntropyIndicesandtheTheilIndices. 70 5.4.3 DecomposabilityofCertainIndices...................... 72 5.5 EstimationwithPartialInformation................................ 77 5.5.1 BoundsontheGiniIndex................................. 77 5.5.2 Parameter Identification Using the Mean andtheGiniIndex......................................... 78 5.6 MomentDistributions .............................................. 80 5.7 RelationsBetweenInequalityMeasures........................... 82 5.8 SampleVersionsofAnalyticMeasuresofInequality............. 83 5.8.1 AbsoluteandRelativeMeanDeviationandthe SamplePietraIndex....................................... 83 5.8.2 TheSampleAmatoandBonferroniIndices.............. 85 5.8.3 TheSampleStandardDeviationandCoefficient ofVariation................................................ 86 5.8.4 Gini’sMeanDifference................................... 87 5.8.5 TheSampleGiniIndex ................................... 89 5.8.6 SampleLorenzCurve..................................... 91 5.8.7 BiasoftheSampleLorenzCurveandGiniIndex....... 93 5.8.8 AsymptoticDistributionofLorenzOrdinates andIncomeShares ........................................ 97 5.8.9 TheEltetoandFrigyesIndices ........................... 101 5.8.10 FurtherClassicalSampleMeasuresofInequality ....... 102 5.8.11 TheSampleAtkinsonandGeneralizedEntropyIndices 105 5.8.12 TheKolmInequalityIndices ............................. 108 5.8.13 AdditionalSampleInequalityIndices.................... 108 5.9 ANewClassofInequalityMeasures .............................. 109 5.10 Exercises ............................................................ 111 6 FamiliesofLorenzCurves................................................. 115 6.1 BasicResults........................................................ 115 6.1.1 ACharacterizationoftheLorenzCurve ................. 116 6.1.2 LorenzCurvesofSomeCommonDistributions......... 116 6.1.3 TranslatedandTruncatedLorenzCurves................ 117 6.1.4 TheModalityoftheIncomeDensityFunction .......... 119 6.2 TheAlchemyofLorenzCurves.................................... 120 6.3 ParametricFamiliesofLorenzCurves............................. 122 6.3.1 SomeHierarchicalFamilies .............................. 123 6.3.2 GeneralQuadraticLorenzCurves........................ 124 6.3.3 OtherParametricFamilies................................ 128