Contents

6.1. Introduction 361

6.1.1 WhyStatisticalMethods? 361

6.1.2 BasicNotationandTerminology 363

6.1.3 AGuidetotheChapter 364

6.2. Data 364

6.2.1 Data Sources 364

6.2.1.1 Administrative Data 365

6.2.1.2 SurveyData 365

6.2.2 Data Structure 365

6.2.2.1 Simple Design 365

6.2.2.2 Complex Design 366

6.2.3 Data Problems 366

6.2.3.1 Measurement Error and Data Contamination 366

6.2.3.2 Incomplete Information 367

6.2.4 Grouped Data 368

6.3.

6.3.1 Parametric Estimation 371

6.3.1.1 Pareto 371

6.3.1.2 Lognormal 373

6.3.1.3 Generalized Beta 374

6.3.1.4 GoodnessofFit 375

6.3.2 Kernel Method 377

6.3.2.1 From Histogram to Kernel Estimator 377

6.3.2.2 Bandwidth Selection 380

6.3.2.3 Adaptive Kernel Estimator 381

6.3.2.4 Multivariate and Conditional Density 382

6.3.3 Finite Mixture Models 383

6.3.3.1 A Group Decomposition Approach 383

6.3.3.2 Number of Components and Number of Groups 385

6.3.3.3 Group Profiles Explanation 387

6.3.3.4 Finite Mixture of Regressions 390

6.3.4 Finite Sample Properties 392

6.4. Welfare Indices 394

Handbook of Income Distribution, Volume 2A

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ISSN 1574-0056, http://dx.doi.org/10.1016/B978-0-444-59428-0.00007-2

6.4.1 Basic Cases 394

6.4.2 Asymptotic Inference 396

6.4.2.1 The Influence Function 397

6.4.2.2 Background Results 398

6.4.2.3 QAD Welfare Indices 399

6.4.3 Application: Inequality Measures 400

6.4.3.1 The Generalized Entropy Class 400

6.4.3.2 The Mean Deviation and Its Relatives 403

6.4.3.3 The Gini Coefficient 404

6.4.4 Application: Poverty Measures 407

6.4.4.1 Foster-Greer-Thorbecke (FGT) 408

6.4.4.2 Sen Poverty Index 409

6.4.4.3 Sen-Shorrocks-ThonPovertyIndex 410

6.4.5 FiniteSampleProperties 412

6.4.5.1 AsymptoticandBootstrapMethods 412

6.4.5.2 Simulation Evidence 413

6.4.5.3 Inference with Heavy-Tailed Distributions 414

6.4.5.4 Testing Equality of Inequality Measures 417

6.4.6 Parametric Approaches 421

6.5. Distributional Comparisons 423

6.5.1 Ranking and Dominance: Principles 423

6.5.1.1 Dominance and Welfare Indices 423

6.5.1.2 Stochastic Dominance 425

6.5.2 Ranking and Dominance: Implementation 426

6.5.2.1 Asymptotic Distributions 426

6.5.2.2 Dominance: An Intuitive Application 430

6.5.2.3 The Null Hypothesis: Dominance or Nondominance 432

6.5.2.4 Hypothesis Testing 435

6.6.

6.6.1 Contamination 437

6.6.1.1 The Concept of Robustness 437

6.6.1.2 Robustness, Welfare Indices, and Distributional Comparisons 438

6.6.1.3 Model Estimation 440

6.6.2 Incomplete Data 441

6.6.2.1 Censored and Truncated Data 441

6.6.2.2 Trimmed Data 443

6.6.3 Semiparametric Methods 447

6.6.3.1 The Model 448

6.6.3.2 Model Estimation 448

6.6.3.3 Choice of β 449

6.6.3.4 Inequality and Dominance 449

6.7. Conclusions 452

6.7.1 Important Lessons: A Round-Up 453

6.7.2 A Worked Example 456

6.7.3 A Cribsheet 458

Acknowledgments 458

References 458