The two-sample location shift model under log-concavity

Abstract

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 Vanet al. (1998). 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 use a truncated version of the one step estimator to theoretically demonstrate adaptivity, our simulations indicate that the one step estimators perform best with zero truncation, eliminating the need for tuning during practical implementation. Notably, the efficiency of the truncated one step estimators steadily increases as the truncation level decreases, and those with low levels of truncation exhibit nearly identical empirical performance to the estimator with zero truncation. We apply our method to investigate the location shift in the distribution of Spanish annual household incomes following the 2008 financial crisis.