Two random variables X and Y belong to the same location-scale family if there are constants @m and @s such that Y and @[email protected] have the same distribution. In this paper we consider non-parametric estimation of the parameters @m and @s under minimal assumptions regarding the form of the distribution functions of X and Y. We discuss an approach to the estimation problem that is based on asymptotic likelihood considerations. Our results enable us to provide a methodology that can be implemented easily and which yields estimators that are often near optimal when compared to fully parametric methods. We evaluate the performance of the estimators in a series of Monte Carlo simulations.