An asymmetric area model-based approach for small area estimation applied to survey data

The Birnbaum–Saunders distribution is asymmetrical and has received considerable attention due to its properties and its relationship with the normal distribution. In this paper, we propose a methodology for estimating the mean of small areas based on a Birnbaum–Saunders distribution which is reparameterized in terms of its mean, similarly to the normal distribution, but in an asymmetric framework. In addition, the variance of the reparameterized Birnbaum–Saunders distribution is a function of its mean, similarly to the gamma distribution, which allows a GLM type modeling to be conducted. The Birnbaum–Saunders area model has properties that are unavailable in its competing models, as describing the mean in the original scale, unlike the existing models which employ a logarithmic transformation that reduces the test power and complicates the interpretation of results. The Birnbaum–Saunders area model can be formulated similarly as the Gaussian area model, permitting us to capture the essence of the small area estimation based on sample means and variances obtained from the areas. The methodology includes a formulation based on the Fay–Herriot model, estimation of model parameters with the maximum likelihood and Bayes empirical methods, as well as diagnostics using residuals. We illustrate the methodology with real-world survey data and compare the results with those obtained by the standard Fay–Herriot model.