A COURSE OF SMALL AREA ESTIMATION

Preface
Contents
Acronyms
1 Small Area Estimation
	1.1 Introduction
	1.2 Mixed Models
	1.3 The Data Files
		1.3.1 The LFS Data Files
		1.3.2 The LCS Data Files
	References
2 Design-Based Direct Estimation
	2.1 Introduction
	2.2 Survey Sampling Theory
	2.3 Direct Estimator of the Total and the Mean
	2.4 Estimator of the Ratio
	2.5 Other Direct Estimators of the Mean and the Total
	2.6 Bootstrap Resampling for Variance Estimation
	2.7 Jackknife Resampling for Variance Estimation
		2.7.1 Delete-One-Cluster Jackknife for Estimators of Domain Parameters
	2.8 R Codes for Design-Based Direct Estimators
		2.8.1 Horvitz–Thompson Direct Estimators of the Total and the Mean
		2.8.2 Hájek Direct Estimator of the Mean and the Total
		2.8.3 Jackknife Estimator of Variances
		2.8.4 Functions for Calculating Direct Estimators
	References
3 Design-Based Indirect Estimation
	3.1 Introduction
	3.2 Basic Synthetic Estimator
	3.3 Post-Stratified Estimator
	3.4 Sample Size Dependent Estimator
	3.5 Generalized Regression Estimator
	3.6 Estimators of Unemployment Rates
	3.7 A Labor Force Survey
		3.7.1 Weight Calibration and Benchmarking
		3.7.2 Resampling Methods for the LFS
	3.8 R Codes for Design-Based Indirect Estimators
		3.8.1 Basic Synthetic Estimator of the Total
		3.8.2 Post-stratified Estimator of the Total
		3.8.3 Generalized Regression Estimator of the Mean
	References
4 Prediction Theory
	4.1 Introduction
	4.2 The Predictive Approach
	4.3 Prediction Theory Under the Linear Model
	4.4 The General Prediction Theorem
	4.5 BLUPs for Some Simple Models
	4.6 R Codes for BLUPs
	References
5 Linear Models
	5.1 Introduction
	5.2 Fixed Effects Linear Models
	5.3 Linear Models with One Fixed Factor
	5.4 BLUPs Based on Linear Models with Fixed Effects
		5.4.1 Regression Synthetic Estimator
		5.4.2 Estimators Without Domain Dependent Intercept
		5.4.3 Estimators with Domain Dependent Intercept
	5.5 R Codes for BLUPs
	References
6 Linear Mixed Models
	6.1 Introduction
	6.2 Linear Mixed Models with Known Variances
		6.2.1 Introduction
		6.2.2 Least Squares Estimation of β
		6.2.3 BLUP of a Linear Combination of Effects
	6.3 Linear Mixed Models with Unknown Variances
	6.4 Maximum Likelihood Estimation
		6.4.1 Description of the Method
		6.4.2 Maximum Likelihood Estimators for Alternative Parameters
	6.5 Residual Maximum Likelihood Estimation
		6.5.1 Description of the Method
		6.5.2 REML Estimators for Alternative Parameters
		6.5.3 Further REML Equations for Linear Mixed Models
	6.6 Henderson 3 Estimation
		6.6.1 Description of the Method
		6.6.2 Moments of Henderson 3 Estimators
	6.7 R Codes for Fitting Linear Mixed Models
		6.7.1 Library lme4
		6.7.2 Library nlme
	References
7 Nested Error Regression Models
	7.1 Introduction
	7.2 The NER Model
	7.3 ML Estimators
	7.4 ML Estimators for Alternative Parameters
	7.5 REML Estimators
	7.6 REML Estimators for Alternative Parameters
	7.7 H3 Estimators
	7.8 Moments of H3 Estimators
	7.9 Simulation Experiment
	7.10 R Codes
		7.10.1 MLEs
		7.10.2 Auxiliary Functions
	References
8 EBLUPs Under Nested Error Regression Models
	8.1 Introduction
	8.2 The NER Model
	8.3 BLUP of a Domain Mean
	8.4 EBLUP of a Single Observation
	8.5 Parametric Bootstrap Estimation of MSEs
	8.6 Model-Assisted Estimation
	8.7 Simulation Experiment
		8.7.1 Artificial Population
		8.7.2 Estimators and Performance Measures
		8.7.3 Numerical Results and Conclusions
	8.8 R Codes
		8.8.1 EBLUPs for LFS Data
		8.8.2 EBLUPs and MA Estimators for LCS Data
	References
9 Mean Squared Error of EBLUPs
	9.1 Introduction
	9.2 The MSE of EBLUPs of Model Effects
		9.2.1 All Model Parameters Are Known
		9.2.2 Known Variances and Unknown Regression Parameters
		9.2.3 All Model Parameters Are Unknown
	9.3 The MSE of EBLUPs of Population Linear Parameters
	9.4 Analytic Estimation of the MSE of EBLUPs
	9.5 MSE Approximation in NER Models
	9.6 MSE Estimation in NER Models
		9.6.1 Henderson 3 Estimation of Variance Components
		9.6.2 REML Estimation of Variance Components
		9.6.3 ML Estimation of Variance Components
	9.7 MSE Approximation in Linear Models with One Fixed Factor
	9.8 Simulation Experiment
		9.8.1 Samples
		9.8.2 EBLUPs and MSEs
		9.8.3 Algorithm
	9.9 R Codes for MSEs
	References
10 EBPs Under Nested Error Regression Models
	10.1 Introduction
	10.2 The Conditional Distribution of Normal Vectors
	10.3 The Nested Error Regression Model
	10.4 EBPs of Domain Means
	10.5 EBPs of Additive Parameters
		10.5.1 Poverty Proportion
		10.5.2 Poverty Gap
		10.5.3 Average Income
	10.6 EBPs Under Subdomain-Level NER Models
		10.6.1 Poverty Proportion
		10.6.2 Poverty Gap
		10.6.3 Average Income
	10.7 ELL Predictors of Poverty Indicators
		10.7.1 Poverty Proportion
		10.7.2 Poverty Gap
		10.7.3 Average Income
	10.8 MSE of Empirical Best Predictors
		10.8.1 Case 1
		10.8.2 Case 2
		10.8.3 Case 3
	10.9 R Codes for EBPs
	References
11 EBLUPs Under Two-Fold Nested Error Regression Models
	11.1 Introduction
	11.2 The Two-fold Nested Error Regression Model
	11.3 The Model with Known Variance Components
	11.4 REML Estimators for Alternative Parameters
		11.4.1 Matrix Calculations
	11.5 The Henderson 3 Method
		11.5.1 Calculation of M1
		11.5.2 Calculation of M2
		11.5.3 Calculation of M3
	11.6 EBLUP of a Subdomain Mean
	11.7 Mean Squared Error of the EBLUP of a Subdomain Mean
		11.7.1 Calculation of g1(θ)
		11.7.2 Calculation of g2(θ)
		11.7.3 Calculation of g3(θ)
		11.7.4 Calculation of g4(θ)
	11.8 Simulation Experiments
		11.8.1 Simulation 1
		11.8.2 Simulation 2
	11.9 R Codes for EBLUPs
	References
12 EBPs Under Two-Fold Nested Error Regression Models
	12.1 Introduction
	12.2 Two-fold Nested Error Regression Models
		12.2.1 The Population Model
		12.2.2 The Sample Model
		12.2.3 The Non-sample Model
		12.2.4 The Inverse of the Variance Matrix
	12.3 The Conditional Distribution of yr given ys
		12.3.1 Conditional Mean Vector
		12.3.2 Conditional Covariance Matrix
		12.3.3 Conditional Variances
	12.4 Monte Carlo EBP of an Additive Parameter
		12.4.1 Introduction
		12.4.2 Auxiliary Variables with Finite Number of Values
	12.5 EBPs of Poverty Indicators
		12.5.1 Poverty Proportion
		12.5.2 Poverty Gap
	12.6 EBPs of Average Income Indicators
	12.7 Parametric Bootstrap MSE Estimator
	12.8 R Codes for EBPs
	References
13 Random Regression Coefficient Models
	13.1 Introduction
	13.2 The RRC Model with Covariance Parameters
		13.2.1 The Model
		13.2.2 REML Estimators
		13.2.3 EBLUP of the Domain Mean
	13.3 The RRC Model Without Covariance Parameters
		13.3.1 The Model
		13.3.2 REML Estimators
			13.3.2.1 Matrix Calculations for the RRC Model
		13.3.3 EBLUP of a Domain Mean
		13.3.4 MSE of the EBLUP
			Calculation of g1(θ)
			Calculation of g2(θ)
			Calculation of g3(θ)
			Calculation of g4(θ)
	13.4 R Codes for EBLUPs
	References
14 EBPs Under Unit-Level Logit Mixed Models
	14.1 Introduction
	14.2 The Unit-Level Logit Mixed Model
	14.3 MSM Algorithm
	14.4 EM Algorithm
		14.4.1 Introduction
		14.4.2 EM Algorithm for the Logit Regression Model
	14.5 ML-Laplace Approximation Algorithm
		14.5.1 Introduction
		14.5.2 The Laplace Approximation to the Likelihood
		14.5.3 The AIC
	14.6 Empirical Best Predictors
		14.6.1 EBP of pdj
		14.6.2 EBP of μd and μd
		14.6.3 EBP of ydj
		14.6.4 EBP of Yd
			14.6.4.1 Predictors with Continuous Auxiliary Variables
			14.6.4.2 Predictors with Categorical Auxiliary Variables
	14.7 MSE of Empirical Best Predictors
		14.7.1 Categorical Auxiliary Variables
			Bootstrap Estimation of the MSE of a Predictor of μd
			Bootstrap Estimation of the MSE of a Predictor of Yd
		14.7.2 Continuous Auxiliary Variables
			Bootstrap Estimation of the MSE of a Predictor of μd
			Bootstrap Estimation of the MSE of a Predictor of Yd
			Census File with Unidentified Sample Units
	14.8 R Codes for EBPs
	References
15 EBPs Under Unit-Level Two-Fold Logit Mixed Models
	15.1 Introduction
	15.2 The Model
	15.3 ML-Laplace Approximation Algorithm
		15.3.1 The Laplace Approximation to the Likelihood
		15.3.2 ML-Laplace Algorithm
		15.3.3 Derivatives of Gd
		15.3.4 AIC
	15.4 Empirical Best Predictors
		15.4.1 EBP of pdtj
		15.4.2 EBP of μdt and μdt
		15.4.3 EBP of ydtj
		15.4.4 EBP of Ydt
			15.4.4.1 Predictors with Continuous Auxiliary Variables
			15.4.4.2 Predictors with Categorical Auxiliary Variables
	15.5 MSE of Empirical Best Predictors
		15.5.1 Bootstrap Estimation of the MSE of the EBP of μdt
		15.5.2 Bootstrap Estimation of the MSE of the EBP of Ydt
	15.6 Simulation Experiment
	15.7 R Codes for EBPs
	References
16 Fay–Herriot Models
	16.1 Introduction
	16.2 BLUPs Under Area-Level Linear Mixed Models
	16.3 The Area-Level Fay–Herriot Model
	16.4 Sampling Error Variances
	16.5 Estimation of Model Parameters
		16.5.1 Prasad–Rao Estimator
		16.5.2 Henderson 3 Estimator
		16.5.3 Maximum Likelihood Method
		16.5.4 Residual Maximum Likelihood Method
	16.6 MSE of the EBLUP
		16.6.1 Parametric Bootstrap
	16.7 Bayesian Prediction
		16.7.1 Unknown σu2
	16.8 Selection of Variables
		16.8.1 Transformation of the Target Variable
		16.8.2 Selection of Auxiliary Variables
	16.9 R Codes for EBLUPs
	References
17 Area-Level Temporal Linear Mixed Models
	17.1 Introduction
	17.2 Area-Level Model with Independent Time Effects
		17.2.1 The Model
		17.2.2 Residual Maximum Likelihood Estimation
		17.2.3 EBLUP and Mean Squared Error
			Calculation of g1(θ)
			Calculation of g2(θ)
			Calculation of g3(θ)
			Parametric Bootstrap
		17.2.4 Simulations
	17.3 Area-Level Model with Correlated Time Effects
		17.3.1 The Model
		17.3.2 Residual Maximum Likelihood Estimation
		17.3.3 EBLUP and Mean Squared Error
			Calculation of g1(θ)
			Calculation of g2(θ)
			Calculation of g3(θ)
			Parametric Bootstrap
		17.3.4 Simulations
	17.4 R Codes for EBLUPs
	References
18 Area-Level Spatio-Temporal Linear Mixed Models
	18.1 Introduction
	18.2 Area-Level Spatial Linear Mixed Model
		18.2.1 The Model
		18.2.2 Fitting Methods Based on the Likelihood
		18.2.3 Parametric Bootstrap Estimation of the MSE
	18.3 Area-Level Spatio-Temporal Linear Mixed Model 1
		18.3.1 The Model
		18.3.2 Residual Maximum Likelihood Estimation
		18.3.3 Simulations
	18.4 Area-Level Spatio-Temporal Linear Mixed Model 2
		18.4.1 The Model
		18.4.2 Residual Maximum Likelihood Estimation
		18.4.3 Simulations
	18.5 R Codes for EBLUPs
	References
19 Area-Level Bivariate Linear Mixed Models
	19.1 Introduction
	19.2 The Bivariate Fay–Herriot Model
	19.3 Properties of the BLUPs
	19.4 Maximum Likelihood Estimation
	19.5 Residual Maximum Likelihood Estimation
	19.6 The Matrix of Mean Squared Crossed Errors
	19.7 Auxiliary Results
	19.8 Simulations
		Simulation 1
		Simulation 2
		Simulation 3
	19.9 R Codes for EBLUPs
		19.9.1 Main Program
		19.9.2 R Functions for the BFH Model
	References
20 Area-Level Poisson Mixed Models
	20.1 Introduction
	20.2 The Model
	20.3 MM Algorithm
	20.4 EM Algorithm
	20.5 ML-Laplace Approximation Algorithm
	20.6 PQL Algorithm
	20.7 Empirical Best Predictors
	20.8 MSE of the EBP
		20.8.1 Approximation of the MSE
		20.8.2 Analytic Estimation of the MSE for MM Estimators
		20.8.3 Bootstrap Estimation of the MSE
	20.9 R Codes for EBPs
	References
21 Area-Level Temporal Poisson Mixed Models
	21.1 Introduction
	21.2 The Model with Independent Time Effects
	21.3 ML-Laplace Approximation Algorithm
	21.4 Empirical Best Predictors
		21.4.1 Bootstrap Estimation of the MSE
	21.5 Simulation Experiment
	21.6 R Codes for EBPs
	References
A Some Useful Formulas
Index