Bayesian Estimation of Log-Normal Means with Finite Quadratic Expected Loss

The log-normal distribution is a popular model in biostatistics and other fields of statistics. Bayesian inference on the mean and median of the distribution is problematic because, for many popular choices of the prior for the variance (on the log-scale) parameter, the posterior distribution has no finite moments, leading to Bayes estimators with infinite expected loss for the most common choices of the loss function. We propose a generalized inverse Gaussian prior for the variance parameter, that leads to a log-generalized hyperbolic posterior, for which it is easy to calculate quantiles and moments, provided that they exist. We derive the constraints on the prior parameters that yield finite posterior moments of order r. We investigate the choice of prior parameters leading to Bayes estimators with optimal frequentist mean square error. For the estimation of the lognormal mean we show, using simulation, that the Bayes estimator under quadratic loss compares favorably in terms of frequentist mean square error to known estimators.