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Root n bandwidths selectors in multivariate kernel density estimation
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  • Published: 29 April 2004

Root n bandwidths selectors in multivariate kernel density estimation

  • Tiee-Jian Wu1 &
  • Min-Hsiao Tsai1 

Probability Theory and Related Fields volume 129, pages 537–558 (2004)Cite this article

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Abstract.

Based on a random sample of size n from an unknown d-dimensional density f, the problem of selecting the bandwidths in kernel estimation of f is investigated. The optimal root n relative convergence rate for bandwidth selection is established and the information bounds in this convergence are given, and a stabilized bandwidth selector (SBS) is proposed. It is known that for all d the bandwidths selected by the least squares cross-validation (LSCV) have large sample variations. The proposed SBS, as an improvement of LSCV, will reduce the variation of LSCV without significantly inflating its bias. The key idea of the SBS is to modify the d-dimensional sample characteristic function beyond some cut-off frequency in estimating the integrated squared bias. It is shown that for all d and sufficiently smooth f and kernel, if the bandwidth in each coordinate direction varies freely, then the multivariate SBS is asymptotically normal with the optimal root n relative convergence rate and achieves the (conjectured) ‘‘lower bound’’ on the covariance matrix.

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Authors and Affiliations

  1. Department of Statistics, National Cheng-Kung University, Tainan, 70101, Taiwan

    Tiee-Jian Wu & Min-Hsiao Tsai

Authors
  1. Tiee-Jian Wu
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  2. Min-Hsiao Tsai
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Additional information

Part of the research was done while the first author was visiting the Institute of Statistical Science, Academia Sinica, Taipei, Taiwan. This work was supported by grant NSC-89-2118-M-006-011, NSC-90-2118-M-006-013 and NSC-91-2118-M-006-005 of National Science Council of Taiwan, R.O.C.

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Wu, TJ., Tsai, MH. Root n bandwidths selectors in multivariate kernel density estimation. Probab. Theory Relat. Fields 129, 537–558 (2004). https://doi.org/10.1007/s00440-004-0357-8

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  • Received: 06 April 2002

  • Revised: 25 February 2004

  • Published: 29 April 2004

  • Issue Date: August 2004

  • DOI: https://doi.org/10.1007/s00440-004-0357-8

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  • Key words or phrases: 
  • Characteristic function
  • Cross-validation
  • Information bound
  • Multivariate data
  • Relative convergence rate
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