  In this paper, we consider the two-sample location shift model, a classic
semiparametric model introduced by Stein (1956). This model is known for its
adaptive nature, enabling nonparametric estimation with full parametric
efficiency. Existing nonparametric estimators of the location shift often
depend on external tuning parameters, which restricts their practical
applicability (Van der Vaart and Wellner, 2021). We demonstrate that
introducing an additional assumption of log-concavity on the underlying density
can alleviate the need for tuning parameters. We propose a one step estimator
for location shift estimation, utilizing log-concave density estimation
techniques to facilitate tuning-free estimation of the efficient influence
function. While we employ a truncated version of the one step estimator for
theoretical adaptivity, our simulations indicate that the one step estimators
perform best with zero truncation, eliminating the need for tuning during
practical implementation.
