Adaptive estimation in symmetric location model under log-concavity constraint

TL;DR

This paper develops adaptive estimators for the symmetric location parameter under the log-concavity constraint, reducing tuning parameter dependence and demonstrating asymptotic efficiency through simulations.

Contribution

It introduces a tuning parameter free, adaptive one-step estimator under log-concavity and analyzes the MLE in the shape-restricted model.

Findings

Untruncated one-step estimator is asymptotically efficient.

Proposed estimators reduce tuning parameter dependence.

Simulation results support theoretical efficiency.

Abstract

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.