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Pointwise Optimality of Wavelet Density Estimation for Negatively Associated Biased Sample

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  • Renyu Ye

    (State Key Laboratory of Mechanics and Control of Mechanical Structures, Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
    School of Mathematics and Computation Science, Anqing Normal University, Anqing 246133, China)

  • Xinsheng Liu

    (State Key Laboratory of Mechanics and Control of Mechanical Structures, Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China)

  • Yuncai Yu

    (State Key Laboratory of Mechanics and Control of Mechanical Structures, Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China)

Abstract

This paper focuses on the density estimation problem that occurs when the sample is negatively associated and biased. We constructed a block thresholding wavelet estimator to recover the density function from the negatively associated biased sample. The pointwise optimality of this wavelet density estimation is shown as L p ( 1 ≤ p < ∞ ) risks over Besov space. To validate the effectiveness of the block thresholding wavelet method, we provide some examples and implement the numerical simulations. The results indicate that our block thresholding wavelet density estimator is superior in terms of the mean squared error (MSE) when comparing with the nonlinear wavelet density estimator.

Suggested Citation

  • Renyu Ye & Xinsheng Liu & Yuncai Yu, 2020. "Pointwise Optimality of Wavelet Density Estimation for Negatively Associated Biased Sample," Mathematics, MDPI, vol. 8(2), pages 1-12, February.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:2:p:176-:d:315506
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    References listed on IDEAS

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    1. Liang, Han-Ying, 2000. "Complete convergence for weighted sums of negatively associated random variables," Statistics & Probability Letters, Elsevier, vol. 48(4), pages 317-325, July.
    2. Youming Liu & Junlian Xu, 2014. "Wavelet density estimation for negatively associated stratified size-biased sample," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 26(3), pages 537-554, September.
    3. Roussas, G. G., 1994. "Asymptotic Normality of Random Fields of Positively or Negatively Associated Processes," Journal of Multivariate Analysis, Elsevier, vol. 50(1), pages 152-173, July.
    4. Matula, Przemyslaw, 1992. "A note on the almost sure convergence of sums of negatively dependent random variables," Statistics & Probability Letters, Elsevier, vol. 15(3), pages 209-213, October.
    5. Christophe Chesneau & Isha Dewan & Hassan Doosti, 2012. "Wavelet linear density estimation for associated stratified size-biased sample," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(2), pages 429-445.
    6. Chicken, Eric & Cai, T. Tony, 2005. "Block thresholding for density estimation: local and global adaptivity," Journal of Multivariate Analysis, Elsevier, vol. 95(1), pages 76-106, July.
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