Multilateral Wellbeing Comparison in a Many Dimensioned World: Ordering and Ranking Collections of Groups (Global Perspectives on Wealth and Distribution)

This book addresses the disparities that arise when measuring and modeling societal behavior and progress across the social sciences. It looks at why and how different disciplines and even researchers can use the same data and yet come to different conclusions about equality of opportunity, economic and social mobility, poverty and polarization, and conflict and segregation. Because societal behavior and progress exist only in the context of other key aspects, modeling becomes exponentially more complex as more of these aspects are factored into considerations. The content of this book transcends disciplinary boundaries, providing valuable information on measuring and modeling to economists, sociologists, and political scientists who are interested in data-based analysis of pressing social issues. Acknowledgments 7 Contents 8 List of Figures 11 List of Tables 13 Introduction 15 Chapter 1: Measuring the Wellbeing of Groups 19 1.1 Introduction 19 1.2 An Outline of What Follows 21 1.3 Measuring Wellbeing: The Social Welfare Function 24 1.4 Measuring Wellbeing: The Benthamite Tradition 28 1.5 The Pigou-Dalton Principle: “Inequality Is a Bad Thing” 29 1.6 Polarization 30 1.7 Social Exclusion 31 1.8 Equality of Opportunity and Social Mobility 34 1.9 The Rawlsian Principle and the Focus on Poverty 36 1.10 What to Do Now? 36 References 37 Chapter 2: Statistical Matters 40 2.1 Introduction 40 2.2 Probability Distributions 41 Multivariate Considerations 43 Statistical Independence 45 Independence and Random Samples 47 Independence and Groups 47 Measures of Location and Dispersion 48 Means and Variances and the Expectations Operator 48 Location Measures 51 Inequality Measures 51 Some Unit Free Inequality Measures 51 2.3 Parametric and Non-Parametric Distributions 54 An Example of a Discrete Probability Density Function: The Poisson Distribution 54 An Example of Continuous Probability Density Function: The Normal Distribution 56 Multidimensional Considerations 58 A Note of Caution 58 The Normal Distribution and Central Limit Theorems 59 Estimation of Unknown Parameters 60 2.4 Kernel Estimation 60 Non-Parametric Distributions 60 The Kernel Function 61 Choosing the “H” and the Kernel 62 Choosing “H” 63 Least Squares Cross Validation 64 Likelihood Cross Validation 64 A Variable Bandwidth H: The Adaptive Kernel 65 Consistency of the Kernel Estimator 66 2.5 Stochastic Dominance Relations 66 2.6 Comparing Distributions 67 Tests for Similarity of Two Distributions 68 2.7 The Test Inconsistency Problem 70 Test Inconsistency for Smooth Continuous Alternatives 72 Maximizing the Power of a Test 74 References 76 Chapter 3: Complete Orderings: Index Types and the Ambiguity Problem 78 3.1 Chapter Outline 78 3.2 Introduction 79 3.3 Indices for the Level of Wellbeing 81 3.4 Some Unit Free Inequality Measures 82 3.5 Inequality Adjusted Wellbeing Levels 86 3.6 Polarization Measures 89 3.7 Multivariate Polarization Indices 95 Three Measures of Multivariate Bipolarization 96 3.8 Poverty Measurement 99 Multivariate Poverty, Deprivation and Exclusion Indices 100 3.9 Equal Opportunity and Mobility Indices 104 3.10 Exploring the Impact of Ambiguity 106 References 109 Chapter 4: Partial Orderings 113 4.1 Introduction 114 4.2 Stochastic Dominance Criteria 117 Some Preliminaries 117 Stochastic Dominance Relations 120 What Does for Different “I” Imply for Societal Preferences? 121 4.3 On Restricting the Criterion Space 123 4.4 Stochastic Dominance and Inequality Orderings 125 4.5 Stochastic Dominance and Poverty Orderings 126 4.6 Stochastic Dominance and Polarization 127 4.7 The Problem of Ambiguity and Conditions for its Absence 131 The Case of Perfect Segmentation 132 Dealing with Ambiguity When K = 2 133 Restricting the Preference Space Reduces Ambiguity 133 Ambiguity in Inequality Measures 134 4.8 Determination of Ambiguity Groupings: Non-Ambiguity Cuts and Groups 137 4.9 Tools for Ordering Groups and Quantifying their Differences 139 Ordering Groups, the Utopia-Dystopia Index 139 Measures of Discrepancies Between Distributions 141 Multilateral Transvariation 142 A Distributional Gini Coefficient 142 Multivariate Considerations 143 Inference for Multilateral Transvariation and Distributional Gini Coefficients 143 Multilateral Transvariation 144 The Distributional Gini 146 References 148 Chapter 5: Comparing Latent Subgroups 151 5.1 Introduction 151 5.2 Semi-Parametric Mixture Distributions 155 5.3 The Probability of Class Membership of an Agent with an Income x 157 5.4 Estimating the Model 159 5.5 Determining the Number of Classes 160 5.6 Studying the Probability of Class Membership 161 5.7 Comparing the Subgroups 162 5.8 An Example: The Eurozone Income Distribution 162 References 165 Chapter 6: Ambiguity, Comparability, Segmentation and All That 168 6.1 Introduction 168 6.2 An “Absence of Ambiguity” Criteria 173 The Case of Complete Segmentation 173 6.3 Dealing with Ambiguity within Two Groups 175 Restricting the Preference Space Reduces Ambiguity 176 6.4 Two Ambiguity Indices 177 A Leshno–Levy Based Index 177 A Transvariation Based Index 179 6.5 Ambiguity in Inequality Measures 180 Assessing the Extent of Incoherent Ranking 183 6.6 Determination of Ambiguity Groupings: Unambiguous Cuts and Groups 183 6.7 An Empirical Application 185 The Data 185 Exploring the Impact of Ambiguity 186 Partition Analysis 190 6.8 Conclusions 194 References 194 Chapter 7: Some Applications 196 7.1 Introduction 196 7.2 An Example of Canadian Unidimensional Income Distribution Analysis 197 7.3 A Multidimensional Equal Opportunity Example: German Educational Attainment 199 7.4 An Example in Portfolio Choice 202 7.5 Gender Equality in Sub Saharan Africa Irrigation Schemes 204 7.6 A Multidimensional Human Development Example 210 References 214 Index 218 Front Matter ....Pages i-xx Measuring the Wellbeing of Groups (Gordon Anderson)....Pages 1-21 Statistical Matters (Gordon Anderson)....Pages 23-60 Complete Orderings: Index Types and the Ambiguity Problem (Gordon Anderson)....Pages 61-95 Partial Orderings (Gordon Anderson)....Pages 97-134 Comparing Latent Subgroups (Gordon Anderson)....Pages 135-151 Ambiguity, Comparability, Segmentation and All That (Gordon Anderson)....Pages 153-180 Some Applications (Gordon Anderson)....Pages 181-202 Back Matter ....Pages 203-205