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Root \(n\) estimates of vectors of integrated density partial derivative functionals

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Abstract

Based on a random sample of size \(n\) from an unknown \(d\)-dimensional density \(f\), the nonparametric estimations of a single integrated density partial derivative functional as well as a vector of such functionals are considered. These single and vector functionals are important in a number of contexts. The purpose of this paper is to derive the information bounds for such estimations and propose estimates that are asymptotically optimal. The proposed estimates are constructed in the frequency domain using the sample characteristic function. For every \(d\) and sufficiently smooth \(f\), it is shown that the proposed estimates are asymptotically normal, attain the optimal \(O_p(n^{-1/2})\) convergence rate and achieve the (conjectured) information bounds. In simulation studies the superior performances of the proposed estimates are clearly demonstrated.

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Acknowledgments

The authors are grateful to an Associate Editor and two referees for many valuable suggestions which significantly improved the quality and presentation of the paper. In particular, the critical suggestions by the referees of including the study of estimating the vector functionals (3), at both the theoretical- and practical-levels, led the authors to reconstruct the modified procedure of Sect. 2.3 that significantly improved its first version.

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Authors and Affiliations

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

    Tiee-Jian Wu, Chih-Yuan Hsu, Huang-Yu Chen & Hui-Chun Yu

Authors
  1. Tiee-Jian Wu
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  2. Chih-Yuan Hsu
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  3. Huang-Yu Chen
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  4. Hui-Chun Yu
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Correspondence to Tiee-Jian Wu.

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Research supported by a grant from the National Science Council of Taiwan and by the National Center for Theoretical Sciences (South) of Taiwan.

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Wu, TJ., Hsu, CY., Chen, HY. et al. Root \(n\) estimates of vectors of integrated density partial derivative functionals. Ann Inst Stat Math 66, 865–895 (2014). https://doi.org/10.1007/s10463-013-0428-7

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  • DOI: https://doi.org/10.1007/s10463-013-0428-7

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