A note on the Hodges-Lehmann estimator and the logistic distribution

It is well known that the Hodges-Lehmann estimator is asymptotically efficient for the location parameter of the logistic distribution. In this article we give a simple and direct proof that this property also characterizes the logistic between all the symmetric location distributions under mild conditions. Using pseudolikelihood, we also show how to find from the Hodges-Lehmann estimator an asymptotically efficient estimator of the scale parameter of the logistic distribution.