We revisit the problem of estimating the center of symmetry $\theta$ of an unknown symmetric density $f$. Although stone (1975), Eden (1970), and Sacks (1975) constructed adaptive estimators of $\theta$ in this model, their estimators depend on external tuning parameters. In an effort to reduce the burden of tuning parameters, we impose an additional restriction of log-concavity on $f$. We construct truncated one-step estimators which are adaptive under the log-concavity assumption. Our simulations suggest that the untruncated version of the one step estimator, which is tuning parameter free, is also asymptotically efficient. We also study the maximum likelihood estimator (MLE) of $\theta$ in the shape-restricted model.

翻译：我们重新探讨了未知对称密度$f$的对称中心$\theta$的估计问题。尽管Stone（1975）、Eden（1970）和Sacks（1975）在该模型中构建了$\theta$的自适应估计量，但这些估计量依赖于外部调节参数。为减轻对调节参数的依赖，我们在$f$上附加了对数凹性约束。我们构造了截断一步估计量，该估计量在对数凹性假设下具有自适应性。模拟结果表明，无需调节参数的非截断版本一步估计量也具有渐近有效性。此外，我们还研究了形状约束模型中$\theta$的最大似然估计量（MLE）。