This book provides a self-contained introduction of mixed-effects models and small area estimation techniques. In particular, it focuses on both introducing classical theory and reviewing the latest methods. First, basic issues of mixed-effects models, such as parameter estimation, random effects prediction, variable selection, and asymptotic theory, are introduced. Standard mixed-effects models used in small area estimation, known as the Fay-Herriot model and the nested error regression model, are then introduced. Both frequentist and Bayesian approaches are given to compute predictors of small area parameters of interest. For measuring uncertainty of the predictors, several methods to calculate mean squared errors and confidence intervals are discussed. Various advanced approaches using mixed-effects models are introduced, from frequentist to Bayesian approaches. This book is helpful for researchers and graduate students in fields requiring data analysis skills as well as in mathematical statistics.
Author(s): Shonosuke Sugasawa, Tatsuya Kubokawa
Series: SpringerBriefs in Statistics: JSS Research Series in Statistics
Publisher: Springer
Year: 2023
Language: English
Pages: 126
City: Singapore
Preface
Contents
1 Introduction
References
2 General Mixed-Effects Models and BLUP
2.1 Mixed-Effects Models and Examples
2.2 Best Linear Unbiased Predictors
2.3 REML and General Estimating Equations
2.4 Asymptotic Properties
2.5 Proofs of the Asymptotic Results
References
3 Measuring Uncertainty of Predictors
3.1 EBLUP and the Mean Squared Error
3.2 Approximation of the MSE
3.3 Evaluation of the MSE Under Normality
3.4 Estimation of the MSE
3.5 Confidence Intervals
References
4 Basic Mixed-Effects Models for Small Area Estimation
4.1 Basic Area-Level Model
4.1.1 Fay–Herriot Model
4.1.2 Asymptotic Properties of EBLUP
4.2 Basic Unit-Level Models
4.2.1 Nested Error Regression Model
4.2.2 Asymptotic Properties of EBLUP
References
5 Hypothesis Tests and Variable Selection
5.1 Test Procedures for a Linear Hypothesis on Regression Coefficients
5.2 Information Criteria for Variable or Model Selection
References
6 Advanced Theory of Basic Small Area Models
6.1 Adjusted Likelihood Methods
6.1.1 Strictly Positive Estimate of Random Effect Variance
6.1.2 Adjusted Likelihood for Empirical Bayes Confidence Intervals
6.1.3 Adjusted Likelihood for Solving Multiple Small Area Estimation Problems
6.2 Observed Best Prediction
6.3 Robust Methods
6.3.1 Unit-Level Models
6.3.2 Area-Level Models
References
7 Small Area Models for Non-normal Response Variables
7.1 Generalized Linear Mixed Models
7.2 Natural Exponential Families with Conjugate Priors
7.3 Unmatched Sampling and Linking Models
7.4 Models with Data Transformation
7.4.1 Area-Level Models for Positive Values
7.4.2 Area-Level Models for Proportions
7.4.3 Unit-Level Models and Estimating Finite Population Parameters
7.5 Models with Skewed Distributions
References
8 Extensions of Basic Small Area Models
8.1 Flexible Modeling of Random Effects
8.1.1 Uncertainty of the Presence of Random Effects
8.1.2 Modeling Random Effects via Global–Local Shrinkage Priors
8.2 Measurement Errors in Covariates
8.2.1 Measurement Errors in the Fay–Herriot Model
8.2.2 Measurement Errors in the Nested Error Regression Model
8.3 Nonparametric and Semiparametric Modeling
8.4 Modeling Heteroscedastic Variance
8.4.1 Shrinkage Estimation of Sampling Variances
8.4.2 Heteroscedastic Variance in Nested Error Regression Models
References
