Bayesian inference for quantiles and conditional means in log-normal models

The main topic of the thesis is the proper execution of a Bayesian inference if log-normality is assumed for data. In fact, it is known that a particular care is required in this context, since the most common prior distributions for the variance in log scale produce posteriors for the log-normal mean which do not have finite moments. Hence, classical summary measures of the posterior such as expectation and variance cannot be computed for these distributions. The thesis is aimed at proposing solutions to carry out Bayesian inference inside a mathematically coherent framework, focusing on the estimation of two quantities: log-normal quantiles (first part of the thesis) and conditioned expectations under a general log-normal linear mixed model (second part of the thesis). Moreover, in the latter section, a further investigation on a unit-level small area models is presented, considering the problem of estimating the well-known log-transformed Battese, Harter and Fuller model in the hierarchical Bayes context. Once the existence conditions for the moments of the target functionals posterior are proved, new strategies to specify prior distributions are suggested. Then, the frequentist properties of the deduced Bayes estimators and credible intervals are evaluated through accurate simulations studies: it resulted that the proposed methodologies improve the Bayesian estimates under naive prior settings and are satisfactorily competitive with the frequentist solutions available in the literature. To conclude, applications of the developed inferential strategies are illustrated on real datasets. The work is completed by the implementation of an R package named BayesLN which allows the users to easily carry out Bayesian inference for log-normal data.