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2022 Estimation of the global mode of a density: Minimaxity, adaptation, and computational complexity
Ery Arias-Castro, Wanli Qiao, Lin Zheng
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Electron. J. Statist. 16(1): 2774-2795 (2022). DOI: 10.1214/21-EJS1972

Abstract

We consider the estimation of the global mode of a density under some decay rate condition around the global mode. We show that the maximum of a histogram, with proper choice of bandwidth, achieves the minimax rate that we establish for the setting that we consider. This is based on knowledge of the decay rate. To address the situation where the decay rate is unknown, we propose a multiscale variant consisting in the recursive refinement of a histogram. We show that this variant is minimax adaptive. These methods run in linear time, and we prove in an appendix that this is best possible: There is no estimation procedure running in sublinear time that achieves the minimax rate.

Acknowledgments

We are grateful to two anonymous referees for their comments that helped improve the paper. This work was partially supported by an NSF grant (DMS 1821154).

Citation

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Ery Arias-Castro. Wanli Qiao. Lin Zheng. "Estimation of the global mode of a density: Minimaxity, adaptation, and computational complexity." Electron. J. Statist. 16 (1) 2774 - 2795, 2022. https://doi.org/10.1214/21-EJS1972

Information

Received: 1 June 2021; Published: 2022
First available in Project Euclid: 14 April 2022

Digital Object Identifier: 10.1214/21-EJS1972

Keywords: histogram-based estimation , minimax adaptive , mode estimation , multiscale estimation

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