Hierarchical Bayes Analysis of the Lognormal Distribution under Quadratic Loss

The lognormal distribution is a popular model in many fields of statistics. The mean, the mode, the median and other summaries of the lognormal may be written as special cases of θa,b = exp(aξ + bσ2) for different choices of a, b. Bayesian inference on θa,b under quadratic loss is problematic since, for many popular choices of the prior for σ2, it has no finite moments. In this paper we propose a Generalized Inverse Gaussian prior for σ2 that leads to a log-Generalized Hyperbolic posterior for θa,b, a distribution for which it is easy to calculate quantiles and moments, provided that they exist. We derive the constraints on the prior parameters that yields finite posterior moments of order r and closed form expressions for the posterior mean and variance of θa,b. We investigate the choice of prior parameters leading to a posterior mean with optimal frequentist MSE using a second-order 'small argument' approximation to the modified Bessel functions of the third kind. We also explore prior specification when the hypotheses on which the 'small argument' approximation is based do not hold. The relative quadratic loss function has often been used in the literature to summarize the posterior for θa,b. We show that, under our approach, also the Bayes estimator obtained minimizing the relative quadratic loss may be written in closed form. With reference to the estimation of the lognromal mean we show that popular estimators may be obtained as special cases of our posterior mean; we also show, by simulation, that our posterior mean compares favourably to known estimators of the lognromal mean in terms of frequentist MSE. In particular it is more efficient than the Bayes estimator obtained minimizing the relative quadratic loss.