The crossing point of two different distribution functions may be of interest for different reasons. The comparison of two different production processes with respect to failures may be one field of application, since the point of intersection of the corresponding distribution functions may be used for selecting the production process of superior quality. As a consequence, an estimator for the crossing point is needed. In this paper an estimator sequence is proposed by altering an approach that has been developed by Hawkins and Kochar in 1991. Using an approach suggested by Ferger in 2009, strong consistency and asymptotic normality of the proposed estimator sequence are derived by considering the argmax of a rescaled process which is selected as the scaled estimating error of the estimator sequence. Subsequently, weak convergence of this process to a limit process in the Skorokhod space is shown, where this limit argmax will turn out to satisfy a Gaussian distribution. A similar result has been obtained by Hawkins and Kochar in 1991, but by means of a different approach.