Small Area Estimation

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.

"The book 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." (Zentralblatt MATH, 2016)

• List of Figures xvList of Tables xviiForeword to the First Edition xixPreface to the Second Edition xxiiiPreface to the First Edition xxvii1 *Introduction 11.1 What is a Small Area? 11.2 Demand for Small Area Statistics 31.3 Traditional Indirect Estimators 41.4 Small Area Models 41.5 Model-Based Estimation 51.6 Some Examples 61.6.1 Health 61.6.2 Agriculture 71.6.3 Income for Small Places 81.6.4 Poverty Counts 81.6.5 Median Income of Four-Person Families 81.6.6 Poverty Mapping 82 Direct Domain Estimation 92.1 Introduction 92.2 Design-Based Approach 102.3 Estimation of Totals 112.3.1 Design-Unbiased Estimator 112.3.2 Generalized Regression Estimator 132.4 Domain Estimation 162.4.1 Case of No Auxiliary Information 162.4.2 GREG Domain Estimation 172.4.3 Domain-Specific Auxiliary Information 182.5 Modified GREG Estimator 212.6 Design Issues 232.6.1 Minimization of Clustering 242.6.2 Stratification 242.6.3 Sample Allocation 242.6.4 Integration of Surveys 252.6.5 Dual-Frame Surveys 252.6.6 Repeated Surveys 262.7 *Optimal Sample Allocation for Planned Domains 262.7.1 Case (i) 262.7.2 Case (ii) 292.7.3 Two-Way Stratification: Balanced Sampling 312.8 Proofs 322.8.1 Proof of ŶGR(𝐱) = 𝐗 322.8.2 Derivation of Calibration Weights 𝑤∗j 322.8.3 Proof of Y = X^T𝐁^when cj = 𝝂T𝐗j 323 Indirect Domain Estimation 353.1 Introduction 353.2 Synthetic Estimation 363.2.1 No Auxiliary Information 363.2.2 *Area Level Auxiliary Information 363.2.3 *Unit Level Auxiliary Information 373.2.4 Regression-Adjusted Synthetic Estimator 423.2.5 Estimation of MSE 433.2.6 Structure Preserving Estimation 453.2.7 *Generalized SPREE 493.2.8 *Weight-Sharing Methods 533.3 Composite Estimation 573.3.1 Optimal Estimator 573.3.2 Sample-Size-Dependent Estimators 593.4 James–Stein Method 633.4.1 Common Weight 633.4.2 Equal Variances 𝜓i = 𝜓 643.4.3 Estimation of Component MSE 683.4.4 Unequal Variances 𝜓i 703.4.5 Extensions 713.5 Proofs 714 Small Area Models 754.1 Introduction 754.2 Basic Area Level Model 764.3 Basic Unit Level Model 784.4 Extensions: Area Level Models 814.4.1 Multivariate Fay–Herriot Model 814.4.2 Model with Correlated Sampling Errors 824.4.3 Time Series and Cross-Sectional Models 834.4.4 *Spatial Models 864.4.5 Two-Fold Subarea Level Models 884.5 Extensions: Unit Level Models 884.5.1 Multivariate Nested Error Regression Model 884.5.2 Two-Fold Nested Error Regression Model 894.5.3 Two-Level Model 904.5.4 General Linear Mixed Model 914.6 Generalized Linear Mixed Models 924.6.1 Logistic Mixed Models 924.6.2 *Models for Multinomial Counts 934.6.3 Models for Mortality and Disease Rates 934.6.4 Natural Exponential Family Models 944.6.5 *Semi-parametric Mixed Models 955 Empirical Best Linear Unbiased Prediction (EBLUP): Theory 975.1 Introduction 975.2 General Linear Mixed Model 985.2.1 BLUP Estimator 985.2.2 MSE of BLUP 1005.2.3 EBLUP Estimator 1015.2.4 ML and REML Estimators 1025.2.5 MSE of EBLUP 1055.2.6 Estimation of MSE of EBLUP 1065.3 Block Diagonal Covariance Structure 1085.3.1 EBLUP Estimator 1085.3.2 Estimation of MSE 1095.3.3 Extension to Multidimensional Area Parameters 1105.4 *Model Identification and Checking 1115.4.1 Variable Selection 1115.4.2 Model Diagnostics 1145.5 *Software 1185.6 Proofs 1195.6.1 Derivation of BLUP 1195.6.2 Equivalence of BLUP and Best Predictor E(𝐦T𝐯|𝐀T𝐲) 1205.6.3 Derivation of MSE Decomposition (5.2.29) 1216 Empirical Best Linear Unbiased Prediction (EBLUP): Basic Area Level Model 1236.1 EBLUP Estimation 1236.1.1 BLUP Estimator 1246.1.2 Estimation of 𝜎2𝑣 1266.1.3 Relative Efficiency of Estimators of 𝜎2𝑣 1286.1.4 *Applications 1296.2 MSE Estimation 1366.2.1 Unconditional MSE of EBLUP 1366.2.2 MSE for Nonsampled Areas 1396.2.3 *MSE Estimation for Small Area Means 1406.2.4 *Bootstrap MSE Estimation 1416.2.5 *MSE of a Weighted Estimator 1436.2.6 Mean Cross Product Error of Two Estimators 1446.2.7 *Conditional MSE 1446.3 *Robust Estimation in the Presence of Outliers 1466.4 *Practical Issues 1486.4.1 Unknown Sampling Error Variances 1486.4.2 Strictly Positive Estimators of 𝜎2𝑣 1516.4.3 Preliminary Test Estimation 1546.4.4 Covariates Subject to Sampling Errors 1566.4.5 Big Data Covariates 1596.4.6 Benchmarking Methods 1596.4.7 Misspecified Linking Model 1656.5 *Software 1697 Basic Unit Level Model 1737.1 EBLUP Estimation 1737.1.1 BLUP Estimator 1747.1.2 Estimation of 𝜎2𝑣 and 𝜎2e 1777.1.3 *Nonnegligible Sampling Fractions 1787.2 MSE Estimation 1797.2.1 Unconditional MSE of EBLUP 1797.2.2 Unconditional MSE Estimators 1817.2.3 *MSE Estimation: Nonnegligible Sampling Fractions 1827.2.4 *Bootstrap MSE Estimation 1837.3 *Applications 1867.4 *Outlier Robust EBLUP Estimation 1937.4.1 Estimation of Area Means 1937.4.2 MSE Estimation 1987.4.3 Simulation Results 1997.5 *M-Quantile Regression 2007.6 *Practical Issues 2057.6.1 Unknown Heteroscedastic Error Variances 2057.6.2 Pseudo-EBLUP Estimation 2067.6.3 Informative Sampling 2117.6.4 Measurement Error in Area-Level Covariate 2167.6.5 Model Misspecification 2187.6.6 Semi-parametric Nested Error Model: EBLUP 2207.6.7 Semi-parametric Nested Error Model: REBLUP 2247.7 *Software 2277.8 *Proofs 2317.8.1 Derivation of (7.6.17) 2317.8.2 Proof of (7.6.20) 2328 EBLUP: Extensions 2358.1 *Multivariate Fay–Herriot Model 2358.2 Correlated Sampling Errors 2378.3 Time Series and Cross-Sectional Models 2408.3.1 *Rao–Yu Model 2408.3.2 State-Space Models 2438.4 *Spatial Models 2488.5 *Two-Fold Subarea Level Models 2518.6 *Multivariate Nested Error Regression Model 2538.7 Two-Fold Nested Error Regression Model 2548.8 *Two-Level Model 2598.9 *Models for Multinomial Counts 2618.10 *EBLUP for Vectors of Area Proportions 2628.11 *Software 2649 Empirical Bayes (EB) Method 2699.1 Introduction 2699.2 Basic Area Level Model 2709.2.1 EB Estimator 2719.2.2 MSE Estimation 2739.2.3 Approximation to Posterior Variance 2759.2.4 *EB Confidence Intervals 2819.3 Linear Mixed Models 2879.3.1 EB Estimation of 𝜇i = 𝐥iT𝜷 + 𝐦Ti 𝐯i 2879.3.2 MSE Estimation 2889.3.3 Approximations to the Posterior Variance 2889.4 *EB Estimation of General Finite Population Parameters 2899.4.1 BP Estimator Under a Finite Population 2909.4.2 EB Estimation Under the Basic Unit Level Model 2909.4.3 FGT Poverty Measures 2939.4.4 Parametric Bootstrap for MSE Estimation 2949.4.5 ELL Estimation 2959.4.6 Simulation Experiments 2969.5 Binary Data 2989.5.1 *Case of No Covariates 2999.5.2 Models with Covariates 3049.6 Disease Mapping 3089.6.1 Poisson–Gamma Model 3099.6.2 Log-Normal Models 3109.6.3 Extensions 3129.7 *Design-Weighted EB Estimation: Exponential Family Models 3139.8 Triple-Goal Estimation 3159.8.1 Constrained EB 3169.8.2 Histogram 3189.8.3 Ranks 3189.9 Empirical Linear Bayes 3199.9.1 LB Estimation 3199.9.2 Posterior Linearity 3229.10 Constrained LB 3249.11 *Software 3259.12 Proofs 3309.12.1 Proof of (9.2.11) 3309.12.2 Proof of (9.2.30) 3309.12.3 Proof of (9.8.6) 3319.12.4 Proof of (9.9.1) 33110 Hierarchical Bayes (HB) Method 33310.1 Introduction 33310.2 MCMC Methods 33510.2.1 Markov Chain 33510.2.2 Gibbs Sampler 33610.2.3 M–H Within Gibbs 33610.2.4 Posterior Quantities 33710.2.5 Practical Issues 33910.2.6 Model Determination 34210.3 Basic Area Level Model 34710.3.1 Known 𝜎2𝑣 34710.3.2 *Unknown 𝜎2𝑣: Numerical Integration 34810.3.3 Unknown 𝜎2𝑣: Gibbs Sampling 35110.3.4 *Unknown Sampling Variances 𝜓i 35410.3.5 *Spatial Model 35510.4 *Unmatched Sampling and Linking Area Level Models 35610.5 Basic Unit Level Model 36210.5.1 Known 𝜎2𝑣 and 𝜎2e 36210.5.2 Unknown 𝜎2𝑣 and 𝜎2e: Numerical Integration 36310.5.3 Unknown 𝜎2𝑣 and 𝜎2e: Gibbs Sampling 36410.5.4 Pseudo-HB Estimation 36510.6 General ANOVA Model 36810.7 *HB Estimation of General Finite Population Parameters 36910.7.1 HB Estimator under a Finite Population 37010.7.2 Reparameterized Basic Unit Level Model 37010.7.3 HB Estimator of a General Area Parameter 37210.8 Two-Level Models 37410.9 Time Series and Cross-Sectional Models 37710.10 Multivariate Models 38110.10.1 Area Level Model 38110.10.2 Unit Level Model 38210.11 Disease Mapping Models 38310.11.1 Poisson-Gamma Model 38310.11.2 Log-Normal Model 38410.11.3 Two-Level Models 38610.12 *Two-Part Nested Error Model 38810.13 Binary Data 38910.13.1 Beta-Binomial Model 38910.13.2 Logit-Normal Model 39010.13.3 Logistic Linear Mixed Models 39310.14 *Missing Binary Data 39710.15 Natural Exponential Family Models 39810.16 Constrained HB 39910.17 *Approximate HB Inference and Data Cloning 40010.18 Proofs 40210.18.1 Proof of (10.2.26) 40210.18.2 Proof of (10.2.32) 40210.18.3 Proof of (10.3.13)–(10.3.15) 402References 405Author Index 431Subject Index 437