Nonparametric estimation of location and scale parameters

Two random variables XX and YY belong to the same location-scale family if there are constants μμ and σσ such that YY and μ+σXμ+σX have the same distribution. In this paper we consider non-parametric estimation of the parameters μμ and σσ under minimal assumptions regarding the form of the distribution functions of XX and YY. 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.