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