Inference with the lognormal distribution

Abstract

Several estimators of the expectation, median and mode of the lognormal distribution are derived. They have the form exp ( μ ^ + b σ ^ 2 ) , where μ ^ and σ ^ 2 are the usual estimators of the mean and variance of the underlying normal distribution, and b is a suitable constant or function of σ ^ 2 . The estimators aim to be approximately unbiased, efficient, or to have a minimax property in the class of estimators we consider. The small-sample properties of these estimators are assessed by simulations and, when possible, analytically. Some of these estimators of the expectation are far more efficient than the maximum likelihood or the minimum-variance unbiased estimator, even for substantial sample sizes.