Preface to the Second Edition

Small area estimation (SAE) deals with the problem of producing reliable estimates of parameters of interest and the associated measures of uncertainty for subpopulations (areas or domains) of a finite population for which samples of inadequate sizes or no samples are available. Traditional "direct estimates," based only on the area-specific sample data, are not suitable for SAE, and it is necessary to "borrow strength" across related small areas through supplementary information to produce reliable "indirect" estimates for small areas. Indirect model-based estimation methods, based on explicit linking models, are now widely used.

The first edition of Small Area Estimation (Rao 2003a) provided a comprehensive account of model-based methods for SAE up to the end of 2002. It is gratifying to see the enthusiastic reception it has received, as judged by the significant number of citations and the rapid growth in SAE literature over the past 12 years. Demand for reliable small area estimates has also greatly increased worldwide. As an example, the estimation of complex poverty measures at the municipality level is of current interest, and World Bank uses a model-based method, based on simulating multiple censuses, in more than 50 countries worldwide to produce poverty statistics for small areas.

The main aim of the present second edition is to update the first edition by providing a comprehensive account of important theoretical developments from 2003 to 2014. New SAE literature is quite extensive and often involves complex theory to handle model misspecifications and other complexities. We have retained a large portion of the material from the first edition to make the book self-contained, and supplemented it with selected new developments in theory and methods of SAE. Notations and terminology used in the first edition are largely retained. As in the first edition, applications are included throughout the chapters. An added feature of the second edition is the inclusion of sections (Sections *Software, *Software, 7.7, 8.11, and 9.11) describing specific R software for SAE, concretely the R package sae (Molina and Marhuenda 2013; Molina and Marhuenda 2015). These sections include examples of SAE applications using data sets included in the package and provide all the necessary R codes, so that the user can exactly replicate the applications. New sections and old sections with significant changes are indicated by an asterisk in the book. Chapter 3 on "Traditional Demographic Methods" from first edition is deleted partly due to page constraints and the fact that the material is somewhat unrelated to mainstream model-based methods. Also, we have not been able to keep up to date with the new developments in demographic methods.

Chapter 1 introduces basic terminology related to SAE and presents selected important applications as motivating examples. Chapter 2, as in the first edition, presents a concise account of direct estimation of totals or means for small areas and addresses survey design issues that have a bearing on SAE. New Section *Optimal Sample Allocation for Planned Domains deals with optimal sample allocation for planned domains and the estimation of marginal row and column strata means in the presence of two-way stratification. Chapter 3 gives a fairly detailed account of traditional indirect estimation based on implicit linking models. The well-known James-Stein method of composite estimation is also studied in the context of sample survey data. New Section *Generalized SPREE studies generalized structure preserving estimation (GSPREE) based on relaxing some interaction assumptions made in the traditional SPREE, which is often used in practice because it makes fuller use of reliable direct estimates at a higher level to produce synthetic estimates. Another important addition is weight sharing (or splitting) methods studied in Section *Weight-Sharing Methods. The weight-sharing methods produce a two-way table of weights with rows as the units in the full sample and columns as the areas such that the cell weights in each row add up to the original sample weight. Such methods are especially useful in micro-simulation modeling that can involve a large number of variables of interest.

Explicit small area models that account for between-area variability are introduced in Chapter 4 (previous Chapter 5), including linear mixed models and generalized linear mixed models such as logistic linear mixed models with random area effects. The models are classified into two broad categories: (i) area level models that relate the small area means or totals to area level covariates; and (ii) unit level models that relate the unit values of a study variable to unit-specific auxiliary variables. Extensions of the models to handle complex data structures, such as spatial dependence and time series structures, are also considered. New Section *Semi-parametric Mixed Models introduces semi-parametric mixed models, which are studied later. Chapter 5 (previous Chapter 6) studies linear mixed models involving fixed and random effects. It gives general results on empirical best linear-unbiased prediction (EBLUP) and the estimation of mean squared error (MSE) of the EBLUP. A detailed account of model identification and checking for linear mixed models is presented in the new Section *Model Identification and Checking. Available SAS software and R statistical software for linear mixed models are summarized in the new Section *Software. The R package sae specifically designed for SAE is also described.

Chapter 6 of the First Edition provided a detailed account of EBLUP estimation of small area means or totals for the basic area level and unit level models, using the general theory given in Chapter 5. In the past 10 years or so, researchers have done extensive work on those two models, especially addressing problems related to model misspecification and other practical issues. As a result, we decided to split the old Chapter 6 into two new chapters, with Chapter 6 focusing on area level models and Chapter 7 addressing unit level models. New topics covered in Chapter 6 include bootstrap MSE estimation (Section *Bootstrap MSE Estimation) and robust estimation in the presence of outliers (Section *Robust estimation in the presence of outliers). Section *Practical issues deals with practical issues related to the basic area level model. It includes important topics such as covariates subject to sampling errors (Section *Practical issues.4), misspecification of linking models (Section *Practical issues.7), benchmarking of model-based area estimators to ensure agreement with a reliable direct estimate when aggregated (Section *Practical issues.6), and the use of "big data" as possible covariates in area level models (Section *Practical issues.5). Functions of the R package sae designed for estimation under the area level model are described in Section *Software. An example illustrating the use of these functions is provided. New topics introduced in Chapter 7 include bootstrap MSE estimation (Section *Bootstrap MSE Estimation), outlier robust EBLUP estimation (Section *Outlier Robust EBLUP Estimation), and M-quantile regression (Section *M-Quantile Regression). Section *Practical Issues deals with practical issues related to the basic unit level model. It presents methods to deal with important topics, including measurement errors in covariates (Section *Practical Issues.4), model misspecification (Section *Practical Issues.5), and semi-parametric nested error models (Sections Semi-parametric Nested Error Model: EBLUP and Semi-parametric Nested Error Model: REBLUP). Most of the published literature assumes that the assumed model for the population values also holds for the sample. However, in many applications, this assumption may not be true due to informative sampling leading to sample selection bias. Section *Practical Issues.3 gives a detailed treatment of methods to make valid inferences under informative sampling. Functions of R package sae dealing with the basic unit level model are described in Section *Software. The use of these functions is illustrated through an application to the County Crop Areas data of Battese, Harter, and Fuller (1988). This application includes calculation of model diagnostics and drawing residual plots. Several important applications are also presented in Chapters 6 and 7.

New chapters 8, 9, and 10 cover the same material as the corresponding chapters in the first edition. Chapter 8 contains EBLUP theory for various extensions of the basic area level and unit level models, providing updates to the sections in the first edition, in particular a more detailed account of spatial and two-level models. Section *Spatial Models on spatial models is updated, and functions of the R package sae dealing with spatial area level models are described in Section *Software. An example illustrating the use of these functions is provided. Section *Two-fold Subarea Level Models presents theory for two-fold subarea level models, which are natural extensions of the basic area level models. Chapter 9 presents empirical Bayes (EB) estimation. The EB method (also called empirical best) is more generally applicable than the EBLUP method. New Section *EB Confidence Intervals gives an account of methods for constructing confidence intervals in the case of basic area level model. EB estimation of general area parameters is the theme of Section *EB Estimation of General Finite Population...