Bayesian inference for quantiles of the log-normal distribution

The log-normal distribution is very popular for modeling positive right-skewed data and represents a common distributional assumption in many environmental applications. Here we consider the estimation of quantiles of this distribution from a Bayesian perspective. We show that the prior on the variance of the log of the variable is relevant for the properties of the posterior distribution of quantiles. Popular choices for this prior, such as the inverse gamma, lead to posteriors without finite moments. We propose the generalized inverse Gaussian and show that a restriction on the choice of one of its parameters guarantees the existence of posterior moments up to a prespecified order. In small samples, a careful choice of the prior parameters leads to point and interval estimators of the quantiles with good frequentist properties, outperforming those currently suggested by the frequentist literature. Finally, two real examples from environmental monitoring and occupational health frameworks highlight the improvements of our methodology, especially in a small sample situation.