Multilateral Wellbeing Comparison in a Many Dimensioned World (eBook)

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