It is now widely recognized that from the point of view of robustness, nonparametric tests, such as the Wilcoxon or normal scores test, should be used in practice for the two-sample location problem instead of the classical $t$-test. But until recently there were no robust estimates for the difference between the two location parameters. In a recent paper [4] Hodges and Lehmann proposed a solution for this problem. After this paper it is clear that all the arguments that can be used in favor of the Wilcoxon test as against the classical $t$-test can be used in favor of the estimate $\operatorname{med} (Y_j - X_i)$ for the difference between the locations as against the classical estimate $(\bar Y - \bar X)$. In their paper quoted above, Hodges and Lehmann propose a whole class of nonparametric estimates corresponding to a class of nonparametric tests both for the two-sample and the one-sample location problems. To indicate this correspondence suppose $h(X, Y)$ is a test-statistic, nonparametric or otherwise for the equality of the locations of $X$ and $Y$. After having observed $(X, Y)$ we estimate the difference between the two location parameters by the amount of shift required to match the samples $X$ and $Y$ in such a way that $h(X, Y)$ is close to its expected value when the shift is zero. For a more precise definition of the estimates the reader is referred to (1.2) below. For a corresponding definition of the one-sample estimates the reader is referred to (3.3). Since the difference between the two one-sample estimates for location is an estimate for the difference between the two location parameters in the two-sample problem the paper of Hodges and Lehmann throws open a whole class of estimates for location in the two-sample problem. The aim of this paper is two-fold. First, how do these estimates compare among themselves in the Behrens-Fisher situation where the scale parameters of the populations can possibly differ? Second, how do these estimates compare with the classical estimate when the scales differ? A basic requirement to be able to answer the above questions is the asymptotic normality of the estimates in the Behrens-Fisher situation and this is shown under fairly general conditions in Sections 2 and 3. In answer to the first question the following main result is proved: If the $\Psi^\ast$-score test is the best linear rank order test when the underlying distribution is $F^\ast$ then the difference between the two one-sample estimates based on the $\Psi^\ast$-score test is more efficient than the simultaneous two-sample estimate based on the $\Psi^\ast$-score test. It follows as a particular case that the difference between the two one-sample estimates based on the normal scores test is more efficient than the simultaneous two-sample estimate based on the normal scores test when the prototype distribution is normal. In answer to the second question, the following result is shown. The differences between the one-sample Hodges-Lehmann estimates has all the advantages over the classical estimate in the case of inequality of variances as in the case of equality of variances. For a more precise statement the reader is referred to the end of Section 4.