Small Area Estimation

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 YGR(x) = X, 32 2.8.2 Derivation of Calibration Weights w* j , 32 2.8.3 Proof of Y = XTB when cj = vT&Xj, 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 psi i = psi, 64 3.4.3 Estimation of Component MSE, 68 3.4.4 Unequal Variances psi 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(mTv;|ATy), 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 sigma² v, 126 6.1.3 Relative Efficiency of Estimators of sigma² v, 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 sigma² v, 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 sigma² v and sigma² 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 my i = I iT beta + m iT v 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 sigma²v , 347 10.3.2 *Unknown sigma²v: Numerical Integration, 348 10.3.3 Unknown sigma²v: Gibbs Sampling, 351 10.3.4 *Unknown Sampling Variances psi 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 sigma²v and sigma²e , 362 10.5.2 Unknown sigma²v and sigma²e: Numerical Integration, 363 10.5.3 Unknown sigma²v and sigma²e: 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