Abstract
The high cost of collecting rich information using large-scale surveys and the abundance of available auxiliary data present researchers with new opportunities to explore estimation methods that extend beyond traditional survey-based estimation. Small area estimation has been an active area of research for decades, resulting in approaches that rely on combining data from multiple sources to tackle estimation at levels of aggregation that are finer than those for which the surveys are designed for. Varying across fields, the term “area” may refer to geography (e.g., state, county, municipality), demographic group (e.g., age, race, ethnicity), or other subpopulations of interest. The term “small” refers to the number of observations available in the area.
Across the world, government agencies have been adopting small area estimation approaches to produce reliable official statistics that are used for effective planning and apportionment of funds. For example, in the United States (U.S.), official statistics of number and percentage of children from low-income families constructed using small area estimation models, as part of the Census Bureau’s Small Area Income and Poverty Estimates (SAIPE) program (Bell et al. 2016), are used to allocate funding under Title I of the Elementary and Secondary Education Act (Census Bureau 2026). In addition to SAIPE, the U.S. Census Bureau implemented small area estimation models in the production of health insurance official statistics for the Small Area Health Insurance Estimates (SAHIE) program (Bauder et al. 2018). Other examples of government programs using small area estimation models are provided in the next paragraph.
The U.S. Bureau of Labor Statistics adopted small area estimation models for the production of state and metropolitan area employment, hours, and earnings official statistics based on the Current Employment Statistics survey (Savitsky and Gershunskaya 2023). The U.S. National Agricultural Statistics Service considered small area estimation models for the production of official statistics for crops (Erciulescu et al. 2019), cash rents (Erciulescu et al. 2018), and agricultural labor (Erciulescu 2018). Small area estimation models were employed for the production of official adult proficiency measures statistics, at the U.S. National Center for Education Statistics (Krenzke et al. 2024) and the United Kingdom’s former Department for Business Innovation & Skills (Gibson and Hewson 2011). Statistics Canada experimented with small area estimation models for the production of official statistics for health characteristics, under-coverage in the census, manufacturing sales, unemployment rates, and employment counts (Hidiroglou et al. 2019). The Chilean Ministerio de Desarrollo (Casas Cordero Valencia et al. 2016) and the World Bank (Elbers et al. 2003) used small area estimation models for poverty mapping.
Statistical models embody assumptions and small area estimation models are no exception to this. When the assumptions are violated, the robustness of the small area estimator is questionable. In addition, the presence of outliers may impact the small area estimation results. Using techniques such as Observed Best Prediction (OBP), Compromised Best Prediction (CBP), heavy-tailed distributions, mixture of distributions, mixture of model specifications, outlier-resistant functions, M-quantile regression, robust small area estimation protects against outliers and misspecification.
With “Robust Small Area Estimation: Methods, Theory, Applications, and Open Problems,” Jiming Jiang and J. Sunil Rao, offer researchers and graduate students of statistics, data science, survey methodology, and geography, a set of practical approaches to robust small area estimation. The frequentist inference framework dominates. In its seven chapters, the book includes a general small area estimation overview, small area estimation methods built on weaker assumptions, outlier robustness, OBP, more flexible models, model selection and diagnostics, and other related topics.
The small area estimation overview chapter starts with direct estimation, design-based indirect estimation, and regression estimation. Then, the two main classes of small area estimation models are presented, area-level models (Fay and Herriot 1979), and unit-level models (Battese et al. 1988), followed by other types of small area estimation models. Direct estimates are not reliable for areas with small sample sizes, and they are not available for areas with zero sample size. By borrowing strength across areas, and from auxiliary information, and/or over space, and/or across time, small area estimation models provide tools to improve the reliability of the direct estimates.
Mixed model prediction and uncertainty estimation, in the context of small area estimation, are then discussed in the small area estimation overview chapter. This material helps the reader familiarize with best predictor (BP) and best linear unbiased predictor (BLUP, which involves normality assumption), functions of unknown model parameters, and their computable versions, empirical best predictor (EBP) and empirical best linear unbiased predictor (EBLUP), functions of estimated model parameters, and mean squared prediction error (MSPE). Reviews, monographs, and software, and then covered at the end of this first chapter.
Chapters 2 and 3 start introducing the robust small area estimation topic, with the question “Assumption or no assumption?” Robustness against violating model assumptions, such as normality or mean and variance functions, is detailed in the second chapter. Starting with the classic least squares (LS) and generalized estimating equations (GEE), the authors then dive into empirical Bayes (EB) and regression average (RA), as approaches to produce weaker-assumption small area estimators. After a slight deviation from the topic, covering outlier robustness (which is then covered to a greater extent in the third chapter), the authors include a discussion of the mixture-model approach for the sampling errors in the unit-level model specification, and the heteroscedastic unit-level model specification. The last topic covered in the second chapter is the estimation of MSPE under weaker model assumptions.
Outlier robustness starts being addressed in the second chapter and then expands in the third chapter. For this, the authors review the use of heavy-tailed distributions (e.g., t-distribution with low degree of freedom, Laplace distribution) in chapter 2 and then start chapter 3 with a discussion on EBLUP. Somewhat of an outlier itself, due to lacking the presence of random (area) effects, quantile regression is covered as an approach for outlier-robust inference. The third chapter concludes with density power divergence and related topics, and a discussion on characterizing the outliers motivated by representative outliers (representative of the non-sampled part of the population).
Robustness to model misspecification is covered in detail, spanning across chapters 4, 5, and 6. Motivated by the need to develop a model fit approach for a model specification that is not subject to change, OMP is detailed in chapter 4. The difference between EBLUP and OBP comes from the estimation goal, full area prediction by minimizing the MSPE over the actual data, as in BLUP (which can be interpreted as a hybrid of BP and the maximum likelihood estimator), versus mixed effect prediction by minimizing the MSPE over all possible predictors, as in OBP. After details on the estimation of the unknown model parameters, under the area-level and unit-level models, the fourth chapter concludes with extensions of OBP, for example, observed best selective prediction, combining covariate selection and estimation of model parameters in a single operation, and compromised best predictor, combining the BLUP and the OBP.
Flexible model specification, and selection and diagnostics are presented as alternatives to OMP, to tackle robustness to model misspecification, in chapters 5 and 6, respectively. For this, semiparametric, nonparametric, and functional small area estimation models are reviewed in the fifth chapter. A common feature of most of these approaches is the approximation of an unknown nonparametric function by a spline function. After an overview of model selection methods (e.g., Mallow’s Cp, information criteria, fence, shrinkage operators), the authors dive deeper into the selection of nonparametric small area estimation models, shrinkage and predictive model selection, variable selection, informal model checking, goodness-of-fit tests, and tailoring.
Robustness to both outliers and model misspecification is briefly mentioned at the end of chapter 7, after an overview of other small area estimation topics, such as benchmarking, Bayesian methods, machine learning methods, missing data, and classified mixed model prediction. By forcing the small area estimation model-based predictors to aggregated to reliable design-based predictors, benchmarking may be interpreted as an approach for robust inference. While Bayesian methods help produce robust small area predictors, the authors could have considered presenting them more in depth, giving similar emphasis as to that given to the frequentist methods. Instead of considering these methods as “other small area estimation topic,” the reader should know that either frequentist inference or Bayesian inference may be used in small area estimation. An important advantage of the Bayesian method is that full (posterior) distribution is produced for the small area parameters, allowing for flexible summarization (e.g., posterior means, posterior variances, posterior credible intervals, posterior probabilities given parameter thresholds). In addition, functions of small area parameters (e.g., ratios, sums) could be predicted straightforwardly using Bayesian methods, and the posterior mean of the random effects variance will be positive (unlike with some frequentist methods, which may lead to negative or zero estimated random effects variance).
Closely related to robustness to model misspecification, machine learning methods can help produce more accurate small area predictions, while benefiting from faster training times for large datasets. In addition to robustness to departures from missing data assumptions, perhaps one should also think about robustness to the contamination of small area estimation with imputation errors. Finally, classified mixed model prediction would tackle robust prediction for out-of-sample areas, by classifying a random effect associated with in-sample areas to the out-of-sample data.
The book concludes with a note that research has just started on robustness to the contamination with differential privacy errors. Adding to this note, perhaps research is also needed on robustness to the contamination with linkage errors and on robustness to informative sampling. Linkage errors arise from probabilistically linking records from different databases. Some early work in this area can be found in Han and Lahiri (2019). Informative sampling refers to scenarios when the response variables are correlated with the sample selection variables. To avoid biased small area estimators, it is critical to capture the sample design in the small area estimation model, say by including the survey weights or the design variables used to construct the survey weights. A recent review of work in this area is provided in Parker et al. (2023). Sustained collaboration between researchers in government agencies, industry, and academia may significantly advance progress in these challenging research areas.
In conclusion, robust small area estimation is presented from various angles, and at various depths, in this book. The mentions of statistical software and rich use of real-data examples supplement the methods and theory, helping intrigue the reader to replicate them, gaining hands-on experience and deeper understanding of the field. Deep dives into explored challenges, ad-hoc questions, and mention of open problems help compel the reader to think deeply.
