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Mathematics > Statistics Theory

arXiv:2105.04287 (math)
[Submitted on 10 May 2021 (v1), last revised 1 Dec 2023 (this version, v2)]

Title:Adaptive estimation in symmetric location model under log-concavity constraint

Authors:Nilanjana Laha
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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.
Comments: In this version, several minor mistakes in the previous version have been corrected. You can find a detailed list of these corrections on the last page of this manuscript. To summarize, fixed algebraic mistake in Theorem 1 proof. Corrected (log n)^2 to (log n)^4 in Lemma B.12. Revised proof of Fact 4 using algebraic inequality (Fact 16)
Subjects: Statistics Theory (math.ST)
MSC classes: Primary 62G99, Secondary 62G20
Cite as: arXiv:2105.04287 [math.ST]
  (or arXiv:2105.04287v2 [math.ST] for this version)
  https://doi.org/10.48550/arXiv.2105.04287

Submission history

From: Nilanjana Laha [view email]
[v1] Mon, 10 May 2021 11:58:19 UTC (1,072 KB)
[v2] Fri, 1 Dec 2023 22:46:59 UTC (243 KB)
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