We propose a class of misaligned data models for addressing typical small area estimation (SAE) problems. In particular, in this paper, a Gaussian process version is considered. Our proposal stems from considering that SAE problems can be conceptually related to spatial misalignment issues: in both contexts, areas for which data (or estimates) are needed (``small areas'' in SAE literature or ``target zones'' according to a standard terminology in geography) are different from areas on which data (or estimates) are available (survey planned domains or ``source zones''). With this connection in mind, we extend atom-based models, borrowed from the most recent hierarchical Bayesian literature on spatial misalignment, for originally addressing the typical SAE questions: (i) combining auxiliary covariates which are available on areal partitions non nested with small areas set (misaligned areal regression problem); (ii) providing small area estimates by using planned domains survey estimates (misaligned areal interpolation problem).
Once atom structure has been defined (as product-grid of all the available source zones), the proposed methodology defines a convenient model for the latent characteristic of interest at atom level, hence induces the distribution of it at any target level by aggregation over component atoms. In this paper we consider the characteristic of interest to be a total, specifically a count, and model it at area level as a Poisson variate with mean given by product Np of population N and incidence p, as results over the chosen area. In the process version here addressed, incidence p (besides of depending on auxiliary covariates) is function of a Gaussian process which is assumed for modeling the spatial point pattern over the entire region. Atom counts derive from integrating the point process over atoms. Computation of such a model has required the implementation of tailored simulation-based methods.
A simulation study examines the capability of the proposed model to improve on traditional SAE model estimates. Moreover, the results are compared to those obtained in a related work by the Authors where a CAR version of the atom-based model is alternatively considered.
