Adaptive estimation in symmetric location model under log-concavity constraint

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