Tests for normality versus lognormality

In applied statistics two of the most widely used distributions for continuous random variables are the normal and the lognormal. In this paper we consider the problem of selecting one of these two distributions. Each distribution is allowed to have unknown location and scale parameters and the lognormal has an unknown shape parameter in addition. Upper bounds are developed for the power of any test which is invariant with respect to location and scale transformations, and the modified ratio of maximized likelihoods (MRML) test is shown to have powers very near Co these upper bounds.