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Praise for the First Edition "This pioneering work, in which Rao provides a comprehensive and up-to-date treatment of small area estimation, will become a classic.... I believe that it has the potential to turn small area estimation...into a larger area of importance to both researchers and practitioners."; --Journal of the American Statistical Association Written by two experts in the field, Small Area Estimation, Second Edition provides a comprehensive and up-to-date account of the methods and theory of small area estimation (SAE), particularly indirect estimation based on explicit small…mehr
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Praise for the First Edition "This pioneering work, in which Rao provides a comprehensive and up-to-date treatment of small area estimation, will become a classic.... I believe that it has the potential to turn small area estimation...into a larger area of importance to both researchers and practitioners."; --Journal of the American Statistical Association Written by two experts in the field, Small Area Estimation, Second Edition provides a comprehensive and up-to-date account of the methods and theory of small area estimation (SAE), 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. Emphasizing real data throughout, the Second Edition maintains a self-contained account of crucial theoretical and methodological developments in the field of SAE. The new edition provides extensive accounts of new and updated research, which often involves complex theory to handle model misspecifications and other complexities. In addition to the information on survey design issues and traditional methods employing indirect estimates based on implicit linking models, Small Area Estimation, Second Edition also features: * Additional sections describe an R package for SAE and applications with R data sets that readers can replicate * Numerous examples of SAE applications throughout the book, including recent applications in U.S. Federal programs * New topical coverage on extended design issues, synthetic estimation, further refinements and solutions to the Fay-Herriot area level model, basic unit level models, and spatial and time series models * A discussion of the advantages and limitations of various SAE methods for model selection from data as well as comparisons of estimates derived from models to reliable values obtained from external sources, such as previous census or administrative data Small Area Estimation, Second Edition 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.
Produktdetails
- Produktdetails
- Verlag: John Wiley & Sons / Wiley
- 2nd Revised edition
- Seitenzahl: 480
- Erscheinungstermin: 24. August 2015
- Englisch
- Abmessung: 240mm x 161mm x 30mm
- Gewicht: 877g
- ISBN-13: 9781118735787
- ISBN-10: 1118735781
- Artikelnr.: 42835609
- Verlag: John Wiley & Sons / Wiley
- 2nd Revised edition
- Seitenzahl: 480
- Erscheinungstermin: 24. August 2015
- Englisch
- Abmessung: 240mm x 161mm x 30mm
- Gewicht: 877g
- ISBN-13: 9781118735787
- ISBN-10: 1118735781
- Artikelnr.: 42835609
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.
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
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
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
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