Location estimation for symmetric log-concave densities

We revisit the problem of estimating the center of symmetry $\theta$ of an unknown symmetric density $f$. Although Stone (1975), Van Eden (1970), and Sacks (1975) constructed adaptive estimators of $\theta$ in this model, their estimators depend on tuning parameters. In an effort to circumvent the dependence on tuning parameters, we impose an additional assumption of log-concavity on $f$. We show that in this shape-restricted model, the maximum likelihood estimator (MLE) of $\theta$ exists. We also study some truncated one-step estimators and show that they are $\sqrt{n}-$consistent, and nearly achieve the asymptotic efficiency bound. We also show that the rate of convergence for the MLE is $O_p(n^{-2/5})$. Furthermore, we show that our estimators are robust with respect to the violation of the log-concavity assumption. In fact, we show that the one step estimators are still $\sqrt{n}$-consistent under some mild conditions. These analytical conclusions are supported by simulation studies.