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On the integrated mean squared error of wavelet density estimation for linear processes

Author

Listed:
  • Aleksandr Beknazaryan

    (University of Tyumen)

  • Hailin Sang

    (University of Mississippi)

  • Peter Adamic

    (Laurentian University)

Abstract

Let $$\{X_n: n\in {{\mathbb {N}}}\}$$ { X n : n ∈ N } be a linear process with density function $$f(x)\in L^2({{\mathbb {R}}})$$ f ( x ) ∈ L 2 ( R ) . We study wavelet density estimation of f(x). Under some regular conditions on the characteristic function of innovations, we achieve, based on the number of nonzero coefficients in the linear process, the minimax optimal convergence rate of the integrated mean squared error of density estimation. Considered wavelets have compact support and are twice continuously differentiable. The number of vanishing moments of mother wavelet is proportional to the number of nonzero coefficients in the linear process and to the rate of decay of characteristic function of innovations. Theoretical results are illustrated by simulation studies with innovations following Gaussian, Cauchy and chi-squared distributions.

Suggested Citation

  • Aleksandr Beknazaryan & Hailin Sang & Peter Adamic, 2023. "On the integrated mean squared error of wavelet density estimation for linear processes," Statistical Inference for Stochastic Processes, Springer, vol. 26(2), pages 235-254, July.
  • Handle: RePEc:spr:sistpr:v:26:y:2023:i:2:d:10.1007_s11203-022-09281-9
    DOI: 10.1007/s11203-022-09281-9
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    References listed on IDEAS

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    1. Hailin Sang & Yongli Sang, 2017. "Memory properties of transformations of linear processes," Statistical Inference for Stochastic Processes, Springer, vol. 20(1), pages 79-103, April.
    2. Leblanc, Frédérique, 1996. "Wavelet linear density estimator for a discrete-time stochastic process: Lp-losses," Statistics & Probability Letters, Elsevier, vol. 27(1), pages 71-84, March.
    3. 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.
    4. Mielniczuk, Jan, 1997. "On the asymptotic mean integrated squared error of a kernel density estimator for dependent data," Statistics & Probability Letters, Elsevier, vol. 34(1), pages 53-58, May.
    5. Faÿ, Gilles & Moulines, Eric & Roueff, François & Taqqu, Murad S., 2009. "Estimators of long-memory: Fourier versus wavelets," Journal of Econometrics, Elsevier, vol. 151(2), pages 159-177, August.
    6. Hailin Sang & Yongli Sang & Fangjun Xu, 2018. "Kernel Entropy Estimation for Linear Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 39(4), pages 563-591, July.
    7. Giraitis, Liudas & Koul, Hira L. & Surgailis, Donatas, 1996. "Asymptotic normality of regression estimators with long memory errors," Statistics & Probability Letters, Elsevier, vol. 29(4), pages 317-335, September.
    8. Giné, Evarist & Madych, W.R., 2014. "On wavelet projection kernels and the integrated squared error in density estimation," Statistics & Probability Letters, Elsevier, vol. 91(C), pages 32-40.
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