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Nonparametric Regression with Singular Design

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  • Lu, Zhan-Qian

Abstract

Theories of nonparametric regression are usually based on the assumption that the design density exists. However, in some applications such as those involving high-dimensional or chaotic time series data, the design measure may be singular and may be likely to have a fractal (nonintegral) dimension. In this paper, the popular Nadaraya-Watson estimator is studied under the general setup that the continuity of the design measure is governed by the local or pointwise dimension. It will be shown in the iid setup that the nonparametric regression estimator achieves a convergence rate which is dependent only on the pointwise dimension. The case of time series data is also studied. For the latter case, a new mixing condition is introduced, and an assumption of marginal or joint density is completely avoided. Three examples, a fractal regression and two applications for predicting chaotic time series, are used to illustrate the implications of the obtained results.

Suggested Citation

  • Lu, Zhan-Qian, 1999. "Nonparametric Regression with Singular Design," Journal of Multivariate Analysis, Elsevier, vol. 70(2), pages 177-201, August.
    • Handle: RePEc:eee:jmvana:v:70:y:1999:i:2:p:177-201
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    Cited by:

    1. Zinde-Walsh, Victoria, 2008. "Kernel Estimation When Density May Not Exist," Econometric Theory, Cambridge University Press, vol. 24(3), pages 696-725, June.
    2. Z. Q. John Lu, 2010. "The Elements of Statistical Learning: Data Mining, Inference, and Prediction," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 173(3), pages 693-694, July.
    3. Nengxiang Ling & Shuhe Hu, 2008. "Asymptotic distribution of partitioning estimation and modified partitioning estimation for regression functions," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(4), pages 353-363.

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