In many studies based on a survey involving economic and agricultural variables, it is frequent to deal with a positive skewed distribution for the response and the log-normality assumption might be suitable in analysing these data. Moreover, it is common to apply small area estimation methods in these situations, leading to the need of investigating the performances of the log-normal linear mixed models.

This work is focused on the unit-level small area models and, particularly, it deals with the estimation of the log-transformed Battese, Harter and Fuller model in the hierarchical Bayes (HB) context. In this framework, a key quantity for the estimation is the posterior predictive distribution, which is used to predict the out-of-samples values in the original data scale.

Considering the findings by Fabrizi and Trivisano (2012) about the existence of the posterior moments of the log-normal mean, a careful analysis about the finiteness of the posterior predictive distribution moments of the fitted model is required. In the main result of this work, it is shown that all the posterior moments of the area means do not always exist, under the most common priors for the variances. Among the distributions that could assure the moments finiteness, the use of the generalized inverse Gaussian distribution with a proper specification of the hyperparameters is proposed. Moreover, since the imposed constraints might lead to priors with excessively light tails, with a consequent possible underestimation issue, a strategy for the hyperparameters setting that preserves the balance among the variance components is presented.

The frequentist properties of the developed model are evaluated in a simulation study and are compared to models adopting inadmissible priors (Molina et al., 2014) and to the current proposals in the empirical Bayes (EB) context (Berg and Chandra, 2014; Molina and Martin, 2018). According to the obtained results, the suggested prior specification produces estimates which outperform the other HB solutions and reach the levels of the EB procedure in terms of bias, MSE, but also frequentist coverage and average width of the intervals.