Location estimation for symmetric log-concave densities
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We revisit the problem of estimating the center of symmetry θ of an unknown symmetric density f. Although Stone (1975), Van Eden (1970), and Sacks (1975) constructed adaptive estimators of θ 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 θ exists. We also study some truncated one-step estimators and show that they are √(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 √(n)-consistent under some mild conditions. These analytical conclusions are supported by simulation studies.
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