To target resources and policies where they are most needed, it is essential that policy-makers are provided with reliable socio-demographic indicators on sub-groups. These sub-groups can be defined by regional divisions or by demographic characteristics and are referred to as areas or domains. Information on these domains is usually obtained through surveys, often planned at a higher level, such as the national level. As sample sizes at disaggregated levels may become small or unavailable, estimates based on survey data alone may no longer be considered reliable or may not be available. Increasing the sample size is time consuming and costly. Small area estimation (SAE) methods aim to solve this problem and achieve higher precision. SAE methods enrich information from survey data with data from additional sources and "borrow" strength from other domains . This is done by modeling and linking the survey data with administrative or register data and by using area-specific structures. Auxiliary data are traditionally population data available at the micro or aggregate level that can be used to estimate unit-level models or area-level models.
Due to strict privacy regulations, it is often difficult to obtain these data at the micro level. Therefore, models based on aggregated auxiliary information, such as the Fay-Herriot model and its extensions, are of great interest for obtaining SAE estimators. Despite the problem of small sample sizes at the disaggregated level, surveys often suffer from high non-response. One possible solution to item non-response is multiple imputation (MI), which replaces missing values with multiple plausible values. The missing values and their replacement introduce additional uncertainty into the estimate. Part I focuses on the Fay-Herriot model, where the resulting estimator is a combination of a design-unbiased estimator based only on the survey data (hereafter called the direct estimator) and a synthetic regression component. Solutions are presented to account for the uncertainty introduced by missing values in the SAE estimator using Rubin's rules. Since financial assets and wealth are sensitive topics, surveys on this type of data suffer particularly from item non-response. Chapter 1 focuses on estimating private wealth at the regionally disaggregated level in Germany. Data from the 2010 Household Finance and Consumption Survey (HFCS) are used for this application. In addition to the non-response problem, income and wealth data are often right-skewed, requiring a transformation to fully satisfy the normality assumptions of the model. Therefore, Chapter 1 presents a modified Fay-Herriot approach that incorporates the uncertainty of missing values into the log-transformed direct estimator of a mean. Chapter 2 complements Chapter 1 by presenting a framework that extends the general class of transformed Fay-Herriot models to account for the additional uncertainty due to MI by including it in the direct component and simultaneously in the regression component of the Fay-Herriot estimator. In addition, the uncertainty due to missing values is also included in the mean squared error estimator, which serves as the uncertainty measure. The estimation of a mean, the use of the log transformation for skewed data, and the arcsine transformation for proportions as target indicators are considered. The proposed framework is evaluated for the three cases in a model-based simulation study. To illustrate the methodology, 2017 data from the HFCS for European Union countries are used to estimate the average value of bonds at the national level. The approaches presented in Chapters 1 and 2 contribute to the literature by providing solutions for estimating SAE models in the presence of multiply imputed survey data. In particular, Chapter 2 presents a general approach that can be extended to other indicators.
To obtain the best possible SAE estimator in terms of accuracy and precision, it is important to find the optimal model for the relationship between the target variable and the auxiliary data. The notion of "optimal" can be multifaceted. One way to look at optimality is to find the best transformation of the target variable to fully satisfy model assumptions or to account for nonlinearity. Another perspective is to identify the most important covariates and their relationship to each other and to the target variable. Part II of this dissertation therefore brings together research on optimal transformations and model selection in the context of SAE. Chapter 3 considers both problems simultaneously for linear mixed models (LMM) and proposes a model selection approach for LMM with data-driven transformations. In particular, the conditional Akaike information criterion is adapted by introducing the Jacobian into the criterion to allow comparison of models at different scales. The methodology is evaluated in a simulation experiment comparing different transformations with different underlying true models. Since SAE models are LMMs, this methodology is applied to the unit-level small-area method, the empirical best predictor (EBP), in an application with Mexican survey and census data (ENIGH - National Survey of Household Income and Expenditure) and shows improvements in efficiency when the optimal (linear mixed) model and the transformation parameters are found simultaneously. Chapter 3 bridges the gap between model selection and optimal transformations to satisfy normality assumptions in unit-level SAE models in particular and LMMs in general. Chapter 4 explores the problem of model selection from a different perspective and for area-level data. To model interactions between auxiliary variables and nonlinear relationships between them and the dependent variable, machine learning methods can be a versatile tool. For unit-level SAE models, mixed-effects random forests (MERFs) provide a flexible solution to account for interactions and nonlinear relationships, ensure robustness to outliers, and perform implicit model selection. In Chapter 4, the idea of MERFs is transferred to area-level models and the linear regression synthetic part of the Fay-Herriot model is replaced by a random forest to benefit from the above properties and to provide an alternative modeling approach. Chapter 4 therefore contributes to the literature by proposing a first way to combine area-level SAE models with random forests for mean estimation to allow for interactions, nonlinear relationships, and implicit variable selection. Another advantage of random forest is its non-extrapolation property, i.e. the range of predictions is limited by the lowest and highest observed values. This could help to avoid transformations at the area-level when estimating indicators defined in a fixed range. The standard Fay-Herriot model was originally developed to estimate a mean, and transformations are required when the indicator of interest is, for example, a share or a Gini coefficient. This usually requires the development of appropriate back-transformations and MSE estimators. 5 presents a Fay-Herriot model for estimating logit-transformed Gini coefficients with a bias-corrected back-transformation and a bootstrap MSE estimator. A model-based simulation is performed to show the validity of the methodology, and regionally disaggregated data from Germany are used to illustrate the proposed approach. 5 contributes to the existing literature by providing, from a frequentist perspective, an alternative to the Bayesian area-level model for estimating Gini coefficients using a logit transformation.
