METHODOLOGICAL PAPER ON CONSTRAINED SMALL AREA ESTIMATION

Small area estimation (SAE) has received considerable attention in the recent decades and is becoming increasingly important for a variety of socio-economic and policy purposes (Ghosh and Rao, 1994; Rao, 2003; Pfeffermann, 2013). As remarked by Tzavidis et al. (2018), SAE is a “simultaneous” rather than a “point” estimation problem. Both areaspecific and ensemble properties of a set of small area estimates, such as their dispersion, rank and order statistics, are undoubtedly of interest. This is a distinctive feature of small area statistics in comparison with the national estimate that is a single number. As we shall explain, to accommodate multiple purposes, the set of small area estimates need to be derived in a constrained fashion, hence the term constrained SAE. This methodological report consists of two parts. In the first part, we review and explore the central ideas relevant to constrained SAE, as well as the most common model-based approaches. Formulating a constrained optimisation problem appears to be a more practical approach. In particular, we shall develop an analytic solution, when the constraints include three first- and second-order empirical moments of the small area estimates. The existing approaches in the literature accommodate only one or two of these constraints. However, even this approach may lack compatibility to the reality of survey sampling, where the national and major domain estimates are derived under the inference framework that is not model-based but design-based. In the second part, we develop a new design-based model-assisted approach to SAE, where we extend generalised regression (e.g. Sarndal et al., 1992) from direct estimation for the whole population to indirect estimation of all the area populations. This enables a practical methodology for estimation, which is coherent numerically between the different aggregation levels, as well as conceptually in terms of the inference outlook.