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On rate-optimal nonparametric wavelet regression with long memory moving average errors

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  • Linyuan Li
  • Kewei Lu

Abstract

We consider the wavelet-based estimators of mean regression function with long memory moving average errors and investigate their asymptotic rates of convergence based on thresholding of empirical wavelet coefficients. We show that these estimators achieve nearly optimal minimax convergence rates within a logarithmic term over a large range of Besov function classes $$B^{s}_{p,q}$$ B p , q s . Therefore, in the presence of long memory non-Gaussian moving average noise, wavelet estimators still achieve nearly optimal convergence rates and provide explicitly the extraordinary local adaptability. The theory is illustrated with some numerical examples. Copyright Springer Science+Business Media Dordrecht 2013

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  • Linyuan Li & Kewei Lu, 2013. "On rate-optimal nonparametric wavelet regression with long memory moving average errors," Statistical Inference for Stochastic Processes, Springer, vol. 16(2), pages 127-145, July.
    • Handle: RePEc:spr:sistpr:v:16:y:2013:i:2:p:127-145
    DOI: 10.1007/s11203-013-9081-2
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    References listed on IDEAS

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    1. Hurvich, Clifford & Lang, Gabriel & Soulier, Philippe, 2005. "Estimation of Long Memory in the Presence of a Smooth Nonparametric Trend," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 853-871, September.
    2. Rainer Von Sachs & Brenda Macgibbon, 2000. "Non‐parametric Curve Estimation by Wavelet Thresholding with Locally Stationary Errors," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(3), pages 475-499, September.
    3. Hall, Peter & Hart, Jeffrey D., 1990. "Nonparametric regression with long-range dependence," Stochastic Processes and their Applications, Elsevier, vol. 36(2), pages 339-351, December.
    4. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521785167.
    5. Iain M. Johnstone & Bernard W. Silverman, 1997. "Wavelet Threshold Estimators for Data with Correlated Noise," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(2), pages 319-351.
    6. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521780506.
    7. 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.
    8. Robinson, Peter M., 1997. "Large-sample inference for nonparametric regression with dependent errors," LSE Research Online Documents on Economics 302, London School of Economics and Political Science, LSE Library.
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    Cited by:

    1. Li, Linyuan, 2015. "Nonparametric adaptive density estimation on random fields using wavelet method," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 346-355.

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