Small Area Estimation

Praise for the First Edition

This pioneering work, in which Rao provides a comprehensive and up–to–date treatment of small area estimation, will become a classic...I believe that it has the potential to turn small area estimation...into a larger area of importance to both researchers and practitioners."

– Journal of the American Statistical Association

Written by two experts in the field, Small Area Estimation, Second Edition provides a comprehensive and up–to–date account of the methods and theory of small area estimation (SAE), particularly indirect estimation based on explicit small area linking models. The model–based approach to small area estimation offers several advantages including increased precision, the derivation of "optimal" estimates and associated measures of variability under an assumed model, and the validation of models from the sample data.

Emphasizing real data throughout, the Second Edition maintains a self–contained account of crucial theoretical and methodological developments in the field of SAE. The new edition provides extensive accounts of new and updated research, which often involves complex theory to handle model misspecifications and other complexities. Including information on survey design issues and traditional methods employing indirect estimates based on implicit linking models, Small Area Estimation, Second Edition also features: 

Additional sections describing the use of R code data sets for readers to use when replicating applications Numerous examples of SAE applications throughout each chapter, including recent applications in U.S. Federal programs New topical coverage on extended design issues, synthetic estimation, further refinements and solutions to the Fay–Herriot area level model, basic unit level models, and spatial and time series models A discussion of the advantages and limitations of various SAE methods for model selection from data as well as comparisons of estimates derived from models to reliable values obtained from external sources, such as previous census or administrative data

Small Area Estimation, Second Edition is an excellent reference for practicing statisticians and survey methodologists as well as practitioners interested in learning SAE methods. The Second Edition is also an ideal textbook for graduate–level courses in SAE and reliable small area statistics.

J. N. K. Rao, PhD, is Professor Emeritus and Distinguished Research Professor in the School of Mathematics and Statistics, Carleton University, Ottawa, Canada. He is an editorial advisor for the Wiley Series in Survey Methodology.

Isabel Molina, PhD, is Associate Professor of Statistics at Universidad Carlos III de Madrid, Spain. 

 

 

List of Figures xv

List of Tables xvii

Foreword to the First Edition xix

Preface to the Second Edition xxiii

Preface to the First Edition xxvii

1 ∗Introduction 1

1.1 What is a Small Area? 1

1.2 Demand for Small Area Statistics, 3

1.3 Traditional Indirect Estimators, 4

1.4 Small Area Models, 4

1.5 Model–Based Estimation, 5

1.6 Some Examples, 6

1.6.1 Health, 6

1.6.2 Agriculture, 7

1.6.3 Income for Small Places, 8

1.6.4 Poverty Counts, 8

1.6.5 Median Income of Four–Person Families, 8

1.6.6 Poverty Mapping, 8

2 Direct Domain Estimation 9

2.1 Introduction, 9

2.2 Design–Based Approach, 10

2.3 Estimation of Totals, 11

2.3.1 Design–Unbiased Estimator, 11

2.3.2 Generalized Regression Estimator, 13

2.4 Domain Estimation, 16

2.4.1 Case of No Auxiliary Information, 16

2.4.2 GREG Domain Estimation, 17

2.4.3 Domain–Specific Auxiliary Information, 18

2.5 Modified GREG Estimator, 21

2.6 Design Issues, 23

2.6.1 Minimization of Clustering, 24

2.6.2 Stratification, 24

2.6.3 Sample Allocation, 24

2.6.4 Integration of Surveys, 25

2.6.5 Dual–Frame Surveys, 25

2.6.6 Repeated Surveys, 26

2.7 ∗Optimal Sample Allocation for Planned Domains, 26

2.7.1 Case (i), 26

2.7.2 Case (ii), 29

2.7.3 Two–Way Stratification: Balanced Sampling, 31

2.8 Proofs, 32

2.8.1 Proof of Y GR( ) = , 32

2.8.2 Derivation of Calibration Weights j , 32

2.8.3 Proof of Y = T when cj = T j, 32

3 Indirect Domain Estimation 35

3.1 Introduction, 35

3.2 Synthetic Estimation, 36

3.2.1 No Auxiliary Information, 36

3.2.2 ∗Area Level Auxiliary Information, 36

3.2.3 ∗Unit Level Auxiliary Information, 37

3.2.4 Regression–Adjusted Synthetic Estimator, 42

3.2.5 Estimation of MSE, 43

3.2.6 Structure Preserving Estimation, 45

3.2.7 ∗Generalized SPREE, 49

3.2.8 ∗Weight–Sharing Methods, 53

3.3 Composite Estimation, 57

3.3.1 Optimal Estimator, 57

3.3.2 Sample–Size–Dependent Estimators, 59

3.4 James Stein Method, 63

3.4.1 Common Weight, 63

3.4.2 Equal Variances i = , 64

3.4.3 Estimation of Component MSE, 68

3.4.4 Unequal Variances i, 70

3.4.5 Extensions, 71

3.5 Proofs, 71

4 Small Area Models 75

4.1 Introduction, 75

4.2 Basic Area Level Model, 76

4.3 Basic Unit Level Model, 78

4.4 Extensions: Area Level Models, 81

4.4.1 Multivariate Fay Herriot Model, 81

4.4.2 Model with Correlated Sampling Errors, 82

4.4.3 Time Series and Cross–Sectional Models, 83

4.4.4 ∗Spatial Models, 86

4.4.5 Two–Fold Subarea Level Models, 88

4.5 Extensions: Unit Level Models, 88

4.5.1 Multivariate Nested Error Regression Model, 88

4.5.2 Two–Fold Nested Error Regression Model, 89

4.5.3 Two–Level Model, 90

4.5.4 General Linear Mixed Model, 91

4.6 Generalized Linear Mixed Models, 92

4.6.1 Logistic Mixed Models, 92

4.6.2 ∗Models for Multinomial Counts, 93

4.6.3 Models for Mortality and Disease Rates, 93

4.6.4 Natural Exponential Family Models, 94

4.6.5 ∗Semi–parametric Mixed Models, 95

5 Empirical Best Linear Unbiased Prediction (EBLUP): Theory 97

5.1 Introduction, 97

5.2 General Linear Mixed Model, 98

5.2.1 BLUP Estimator, 98

5.2.2 MSE of BLUP, 100

5.2.3 EBLUP Estimator, 101

5.2.4 ML and REML Estimators, 102

5.2.5 MSE of EBLUP, 105

5.2.6 Estimation of MSE of EBLUP, 106

5.3 Block Diagonal Covariance Structure, 108

5.3.1 EBLUP Estimator, 108

5.3.2 Estimation of MSE, 109

5.3.3 Extension to Multidimensional Area Parameters, 110

5.4 ∗Model Identification and Checking, 111

5.4.1 Variable Selection, 111

5.4.2 Model Diagnostics, 114

5.5 ∗Software, 118

5.6 Proofs, 119

5.6.1 Derivation of BLUP, 119

5.6.2 Equivalence of BLUP and Best Predictor E( T | T ), 120

5.6.3 Derivation of MSE Decomposition (5.2.29), 121

6 Empirical Best Linear Unbiased Prediction (EBLUP): Basic Area Level Model 123

6.1 EBLUP Estimation, 123

6.1.1 BLUP Estimator, 124

6.1.2 Estimation of 2 , 126

6.1.3 Relative Efficiency of Estimators of 2 , 128

6.1.4 ∗Applications, 129

6.2 MSE Estimation, 136

6.2.1 Unconditional MSE of EBLUP, 136

6.2.2 MSE for Nonsampled Areas, 139

6.2.3 ∗MSE Estimation for Small Area Means, 140

6.2.4 ∗Bootstrap MSE Estimation, 141

6.2.5 ∗MSE of a Weighted Estimator, 143

6.2.6 Mean Cross Product Error of Two Estimators, 144

6.2.7 ∗Conditional MSE, 144

6.3 ∗Robust estimation in the presence of outliers, 146

6.4 ∗Practical issues, 148

6.4.1 Unknown Sampling Error Variances, 148

6.4.2 Strictly Positive Estimators of 2 , 151

6.4.3 Preliminary Test Estimation, 154

6.4.4 Covariates Subject to Sampling Errors, 156

6.4.5 Big Data Covariates, 159

6.4.6 Benchmarking Methods, 159

6.4.7 Misspecified Linking Model, 165

6.5 ∗Software, 169

7 Basic Unit Level Model 173

7.1 EBLUP estimation, 173

7.1.1 BLUP Estimator, 174

7.1.2 Estimation of 2 and 2 e , 177

7.1.3 ∗Nonnegligible Sampling Fractions, 178

7.2 MSE Estimation, 179

7.2.1 Unconditional MSE of EBLUP, 179

7.2.2 Unconditional MSE Estimators, 181

7.2.3 ∗MSE Estimation: Nonnegligible Sampling Fractions, 182

7.2.4 ∗Bootstrap MSE Estimation, 183

7.3 ∗Applications, 186

7.4 ∗Outlier Robust EBLUP Estimation, 193

7.4.1 Estimation of Area Means, 193

7.4.2 MSE Estimation, 198

7.4.3 Simulation Results, 199

7.5 ∗M–Quantile Regression, 200

7.6 ∗Practical Issues, 205

7.6.1 Unknown Heteroscedastic Error Variances, 205

7.6.2 Pseudo–EBLUP Estimation, 206

7.6.3 Informative Sampling, 211

7.6.4 Measurement Error in Area–Level Covariate, 216

7.6.5 Model Misspecification, 218

7.6.6 Semi–parametric Nested Error Model: EBLUP, 220

7.6.7 Semi–parametric Nested Error Model: REBLUP, 224

7.7 ∗Software, 227

7.8 ∗Proofs, 231

7.8.1 Derivation of (7.6.17), 231

7.8.2 Proof of (7.6.20), 232

8 EBLUP: Extensions 235

8.1 ∗Multivariate Fay Herriot Model, 235

8.2 Correlated Sampling Errors, 237

8.3 Time Series and Cross–Sectional Models, 240

8.3.1 ∗Rao Yu Model, 240

8.3.2 State–Space Models, 243

8.4 ∗Spatial Models, 248

8.5 ∗Two–fold Subarea Level Models, 251

8.6 ∗Multivariate Nested Error Regression Model, 253

8.7 Two–fold Nested Error Regression Model, 254

8.8 ∗Two–Level Model, 259

8.9 ∗Models for Multinomial Counts, 261

8.10 ∗EBLUP for Vectors of Area Proportions, 262

8.11 ∗Software, 264

9 Empirical Bayes (EB) Method 269

9.1 Introduction, 269

9.2 Basic Area Level Model, 270

9.2.1 EB Estimator, 271

9.2.2 MSE Estimation, 273

9.2.3 Approximation to Posterior Variance, 275

9.2.4 ∗EB Confidence Intervals, 281

9.3 Linear Mixed Models, 287

9.3.1 EB Estimation of i = i T + Ti i, 287

9.3.2 MSE Estimation, 288

9.3.3 Approximations to the Posterior Variance, 288

9.4 ∗EB Estimation of General Finite Population Parameters, 289

9.4.1 BP Estimator Under a Finite Population, 290

9.4.2 EB Estimation Under the Basic Unit Level Model, 290

9.4.3 FGT Poverty Measures, 293

9.4.4 Parametric Bootstrap for MSE Estimation, 294

9.4.5 ELL Estimation, 295

9.4.6 Simulation Experiments, 296

9.5 Binary Data, 298

9.5.1 ∗Case of No Covariates, 299

9.5.2 Models with Covariates, 304

9.6 Disease Mapping, 308

9.6.1 Poisson Gamma Model, 309

9.6.2 Log–normal Models, 310

9.6.3 Extensions, 312

9.7 ∗Design–Weighted EB Estimation: Exponential Family Models, 313

9.8 Triple–goal Estimation, 315

9.8.1 Constrained EB, 316

9.8.2 Histogram, 318

9.8.3 Ranks, 318

9.9 Empirical Linear Bayes, 319

9.9.1 LB Estimation, 319

9.9.2 Posterior Linearity, 322

9.10 Constrained LB, 324

9.11 ∗Software, 325

9.12 Proofs, 330

9.12.1 Proof of (9.2.11), 330

9.12.2 Proof of (9.2.30), 330

9.12.3 Proof of (9.8.6), 331

9.12.4 Proof of (9.9.11), 331

10 Hierarchical Bayes (HB) Method 333

10.1 Introduction, 333

10.2 MCMC Methods, 335

10.2.1 Markov Chain, 335

10.2.2 Gibbs Sampler, 336

10.2.3 M H Within Gibbs, 336

10.2.4 Posterior Quantities, 337

10.2.5 Practical Issues, 339

10.2.6 Model Determination, 342

10.3 Basic Area Level Model, 347

10.3.1 Known 2 , 347

10.3.2 ∗Unknown 2 : Numerical Integration, 348

10.3.3 Unknown 2 : Gibbs Sampling, 351

10.3.4 ∗Unknown Sampling Variances i, 354

10.3.5 ∗Spatial Model, 355

10.4 ∗Unmatched Sampling and Linking Area Level Models, 356

10.5 Basic Unit Level Model, 362

10.5.1 Known 2 and 2 e , 362

10.5.2 Unknown 2 and 2 e : Numerical Integration, 363

10.5.3 Unknown 2 and 2 e : Gibbs Sampling, 364

10.5.4 Pseudo–HB Estimation, 365

10.6 General ANOVA Model, 368

10.7 ∗HB Estimation of General Finite Population Parameters, 369

10.7.1 HB Estimator under a Finite Population, 370

10.7.2 Reparameterized Basic Unit Level Model, 370

10.7.3 HB Estimator of a General Area Parameter, 372

10.8 Two–Level Models, 374

10.9 Time Series and Cross–sectional Models, 377

10.10 Multivariate Models, 381

10.10.1 Area Level Model, 381

10.10.2 Unit Level Model, 382

10.11 Disease Mapping Models, 383

10.11.1 Poisson–Gamma Model, 383

10.11.2 Log–Normal Model, 384

10.11.3 Two–Level Models, 386

10.12 ∗Two–Part Nested Error Model, 388

10.13 Binary Data, 389

10.13.1 Beta–Binomial Model, 389

10.13.2 Logit–Normal Model, 390

10.13.3 Logistic Linear Mixed Models, 393

10.14 ∗Missing Binary Data, 397

10.15 Natural Exponential Family Models, 398

10.16 Constrained HB, 399

10.17 ∗Approximate HB Inference and Data Cloning, 400

10.18 Proofs, 402

10.18.1 Proof of (10.2.26), 402

10.18.2 Proof of (10.2.32), 402

10.18.3 Proof of (10.3.13) (10.3.15), 402

References 405

Author Index 431

Subject Index 437