A point estimator and a set of confidence intervals based on the Kolmogorov-Smirnov statistic are proposed for the shift parameter in the two-sample problem. Asymptotic distibution of the etimator as well as asymptotic bounds for the lengths of the intervals are derived. The two-sample results are then adapted to the one-sample problem to define an estimator and a set of confidence intervals for the center of a symmetric population.
"Estimation of Shift and Center of Symmetry Based on Kolmogorov-Smirnov Statistics." Ann. Statist. 3 (4) 862 - 873, July, 1975. https://doi.org/10.1214/aos/1176343187