Small area estimation / J.N.K. Rao.

Format:

Book

Author/Creator:

Rao, J. N. K., 1937-

Contributor:

Classes of 1883 and 1884 Fund.

Series:

Wiley series in survey methodology

Language:

English

Subjects (All):

Sampling (Statistics).

Estimation theory.

Physical Description:

xxiii, 313 pages : illustrations ; 24 cm.

Place of Publication:

Hoboken, N.J. : John Wiley, [2003]

Summary:

A much-needed guide to reliable small area statistics

The term "small area" denotes any subpopulation for which direct estimates of adequate precision cannot be produced. In recent years, the demand for reliable small area estimates has greatly increased worldwide due to, among other things, their growing use in formulating policies and programs and the allocation of government funds; regional planning; small business decisions; and similar applications.

Small Area Estimation provides a comprehensive account of the methods and theory of small area estimation, 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.

The clear, detailed coverage includes: Basic terminology related to small area estimation Survey design issues and traditional methods employing indirect estimates based on implicit linking models Linear mixed models and generalized linear mixed models Empirical Best Linear Unbiased Prediction (EBLUP), Empirical Bayes (EB) and Hierarchical Bayes (HB) Estimation Model diagnostics Various extensions including binary response and count data through generalized linear mixed models and time series data through linear mixed models that combine cross-sectional and time series features Important applications of SAE including several in U.S. federal programs

Contents:

1.1 What is a Small Area? 1

1.2 Demand for Small Area Statistics 3

1.3 Traditional Indirect Estimators 3

1.4 Small Area Models 4

1.5 Model-Based Estimation 4

1.6 Some Examples 6

2 Direct Domain Estimation 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 15

2.4.1 Case of no Auxiliary Information 15

2.4.2 GREG Estimation 17

2.4.3 Domain-specific Auxiliary Information 17

2.5 Modified Direct Estimators 20

2.6 Design Issues 21

2.7 Proofs 25

2.7.1 Proof of Y[subscript GR](x) = X 25

2.7.2 Derivation of Calibration Weights w*[subscript j] 25

2.7.3 Proof of Y = X[superscript T]B when c[subscript j] = v[superscript T]x[subscript j] 25

3 Traditional Demographic Methods 27

3.2 Symptomatic Accounting Techniques 28

3.2.1 Vital Rates Method 28

3.2.2 Composite Method 30

3.2.3 Component Methods 30

3.2.4 Housing Unit Method 30

3.3 Regression Symptomatic Procedures 31

3.3.1 Ratio Correlation and Difference Correlation Methods 31

3.3.2 Sample Regression Method 33

3.4 Dual-system Estimation of Total Population 37

3.4.1 Dual-system Model 37

3.4.2 Post-enumeration Surveys 39

3.5 Derivation of Average MSEs 42

4 Indirect Domain Estimation 45

4.2 Synthetic Estimation 46

4.2.1 No Auxiliary Information 46

4.2.2 Auxiliary Information Available 46

4.2.3 Regression-adjusted Synthetic Estimator 51

4.2.4 Estimation of MSE 51

4.2.5 Structure Preserving Estimation 53

4.3 Composite Estimation 57

4.3.1 Optimal Estimator 57

4.3.2 Sample Size Dependent Estimators 60

4.4 James-Stein Method 63

4.4.1 Common Weight 63

4.4.2 Equal Variances [psi subscript i] = [psi] 64

4.4.3 Estimation of Component MSE 68

4.4.4 Unequal Variances [psi subscript i] 71

4.4.5 Extensions 71

4.5 Proofs 72

5 Small Area Models 75

5.2 Basic Area Level (Type A) Model 76

5.3 Basic Unit Level (Type B) Model 78

5.4 Extensions: Type A Models 81

5.4.1 Multivariate Fay-Herriot Model 81

5.4.2 Model with Correlated Sampling Errors 82

5.4.3 Time Series and Cross-sectional Models 83

5.4.4 Spatial Models 86

5.5 Extensions: Type B Models 87

5.5.1 Multivariate Nested Error Regression Model 87

5.5.2 Random Error Variance Linear Model 88

5.5.3 Two-fold Nested Error Regression Model 88

5.5.4 Two-level Model 89

5.5.5 General Linear Mixed Model 90

5.6 Generalized Linear Mixed Models 91

5.6.1 Logistic Regression Models 91

5.6.2 Models for Mortality and Disease Rates 92

5.6.3 Exponential Family Models 93

5.6.4 Semi-parametric Models 93

6 Empirical Best Linear Unbiased Prediction: Theory 95

6.2 General Linear Mixed Model 96

6.2.1 BLUP Estimator 96

6.2.2 MSE of BLUP 98

6.2.3 EBLUP Estimator 99

6.2.4 ML and REML Estimators 100

6.2.5 MSE of EBLUP 103

6.2.6 Estimation of MSE of EBLUP 104

6.2.7 Software 105

6.3 Block Diagonal Covariance Structure 107

6.3.1 EBLUP Estimator 107

6.3.2 Estimation of MSE 108

6.3.3 Extension 110

6.3.4 Model Diagnostics 110

6.4 Proofs 112

6.4.1 Derivation of BLUP 112

6.4.2 Equivalence of BLUP and Best Predictor E(m[superscript T]v|A[superscript T]y) 113

6.4.3 Derivation of the Decomposition (6.2.26) 113

7 EBLUP: Basic Models 115

7.1 Basic Area Level Model 115

7.1.1 BLUP Estimator 116

7.1.2 Estimation of [sigma superscript 2 subscript v] 118

7.1.3 Relative Efficiency of Estimators of [sigma superscript 2 subscript v] 120

7.1.5 MSE Estimation 128

7.1.6 Conditional MSE 131

7.1.7 Mean Product Error of Two Estimators 132

7.1.8 Estimation of Small Area Means 133

7.1.9 Weighted Estimator 134

7.2 Basic Unit Level Model 134

7.2.1 BLUP Estimator 135

7.2.2 Estimation of [sigma superscript 2 subscript v] and [sigma superscript 2 subscript e] 138

7.2.3 MSE of EBLUP 139

7.2.4 MSE Estimation 140

7.2.5 Non-negligible Sampling Rates 141

7.2.7 Pseudo-EBLUP Estimation 148

8 EBLUP: Extensions 153

8.1 Multivariate Fay-Herriot Model 153

8.2 Correlated Sampling Errors 155

8.3 Time Series and Cross-sectional Models 158

8.3.1 Rao-Yu Model 158

8.3.2 State Space Models 162

8.4 Spatial Models 168

8.5 Multivariate Nested Error Regression Model 169

8.6 Random Error Variances Linear Model 171

8.7 Two-fold Nested Error Regression Model 172

8.8 Two-level Model 176

9 Empirical Bayes (EB) Method 179

9.2 Basic Area Level Model 180

9.2.1 EB Estimator 181

9.2.2 MSE Estimation 182

9.2.3 Approximation to Posterior Variance 185

9.2.4 EB Confidence Intervals 191

9.3 Linear Mixed Models 194

9.3.1 EB Estimation 194

9.3.2 MSE Estimation 195

9.3.3 Approximations to the Posterior Variance 196

9.4 Binary Data 197

9.4.1 Case of no Covariates 197

9.4.2 Models with Covariates 202

9.5 Disease Mapping 205

9.5.1 Poisson-Gamma Model 206

9.5.2 Log-normal Models 208

9.5.3 Extensions 209

9.6 Triple-goal Estimation 211

9.6.1 Constrained EB 211

9.6.2 Histogram 213

9.6.3 Ranks 213

9.7 Empirical Linear Bayes 214

9.7.1 LB Estimation 214

9.7.2 Posterior Linearity 217

9.8 Constrained LB 219

9.9 Proofs 220

9.9.1 Proof of (9.2.11) 220

9.9.2 Proof of (9.2.30) 221

9.9.3 Proof of (9.6.6) 221

9.9.4 Proof of (9.7.1) 222

10 Hierarchical Bayes (HB) Method 223

10.2 MCMC Methods 224

10.2.1 Markov Chain 224

10.2.2 Gibbs Sampler 225

10.2.3 M-H Within Gibbs 226

10.2.4 Practical Issues 227

10.2.5 Posterior Quantities 230

10.2.6 Model Determination 232

10.3 Basic Area Level Model 237

10.3.1 Known [sigma superscript 2 subscript v] 237

10.3.2 Unknown [sigma superscript 2 subscript v]: Numerical Integration 237

10.3.3 Unknown [sigma superscript 2 subscript v]: Gibbs Sampling 240

10.4 Unmatched Sampling and Linking Area Level Models 243

10.5 Basic Unit Level Model 247

10.5.1 Known [sigma superscript 2 subscript v] and [sigma superscript 2 subscript e] 247

10.5.2 Unknown [sigma superscript 2 subscript v] and [sigma superscript 2 subscript e]: Numerical Integration 247

10.5.3 Unknown [sigma superscript 2 subscript v] and [sigma superscript 2 subscript e]: Gibbs Sampling 248

10.5.4 Pseudo-HB Estimation 251

10.6 General ANOVA Model 254

10.7 Two-level Models 255

10.8 Time Series and Cross-sectional Models 258

10.9 Multivariate Models 263

10.9.1 Area Level Model 263

10.9.2 Unit Level Model 264

10.10 Disease Mapping Models 264

10.10.1 Poisson-gamma Model 264

10.10.2 Log-normal Model 265

10.10.3 Two-level Models 267

10.11 Binary Data 269

10.11.1 Beta-binomial Model 269

10.11.2 Logit-normal Model 270

10.11.3 Logistic Linear Mixed Models 273

10.12 Exponential Family Models 277

10.13 Constrained HB 278

10.14 Proofs 279

10.14.1 Proof of (10.2.26) 279

10.14.2 Proof of (10.2.32) 280

10.14.3 Proof of (10.3.11)-(10.3.13) 280.

Notes:

Includes bibliographical references (pages 283-301) and indexes.

Local Notes:

Acquired for the Penn Libraries with assistance from the Classes of 1883 and 1884 Fund.

ISBN:

0471413747

OCLC:

50643804

Online:

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