Adaptive estimation in symmetric location model under log-concavity constraint

Nilanjana Laha

doi:10.1214/21-ejs1852

Abstract

We revisit the problem of estimating the center of symmetry θ of an unknown symmetric density f. Although Stone (1975), Van Eeden (1970), and Sacks (1975) constructed adaptive estimators of θ 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 indicate 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 θ in the shape-restricted model.

Highlights

In this paper, we revisit the symmetric location model with an additional shaperestriction of log-concavity
Figure 2 implies that Stone and Beran’s estimators have high efficiency when they are equipped with the optimal tuning parameters
When the tuning parameters are non-optimal, the nonparametric estimators suffer in terms of both efficiency and the coverage

Summary

Introduction

We revisit the symmetric location model with an additional shaperestriction of log-concavity. We let P denote the class of all densities on the real line R. For any θ ∈ R, denote by Sθ the class of all densities symmetric about θ. The symmetric location model Ps is given by. Where If is the Fisher information for location. It is well-established that (Huber, 1964, Theorem 3) If is finite if and only if f is an absolutely continuous density satisfying ∞ f (x) f (x)dx < ∞,. −∞ f (x) where f is an L1-derivative of f.

Objectives

Results

Conclusion