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Nonparametric methods for deconvolving multiperiodic functions

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  • Peter Hall
  • Jiying Yin

Abstract

Summary. Multiperiodic functions, or functions that can be represented as finite additive mixtures of periodic functions, arise in problems related to stellar radiation. There they represent the overall variation in radiation intensity with time. The individual periodic components generally correspond to different sources of radiation and have intrinsic physical meaning provided that they can be ‘deconvolved’ from the mixture. We suggest a combination of kernel and orthogonal series methods for performing the deconvolution, and we show how to estimate both the sequence of periods and the periodic functions themselves. We pay particular attention to the issue of identifiability, in a nonparametric sense, of the components. This aspect of the problem is shown to exhibit particularly unusual features, and to have connections to number theory. The matter of rates of convergence of estimators also has links there, although we show that the rate‐of‐convergence problem can be treated from a relatively conventional viewpoint by considering an appropriate prior distribution for the periods.

Suggested Citation

  • Peter Hall & Jiying Yin, 2003. "Nonparametric methods for deconvolving multiperiodic functions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(4), pages 869-886, November.
    • Handle: RePEc:bla:jorssb:v:65:y:2003:i:4:p:869-886
    DOI: 10.1046/j.1369-7412.2003.00420.x
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    Cited by:

    1. Oliver Linton & Michael Vogt, 2012. "Nonparametric estimation of a periodic sequence in the presence of a smooth trend," CeMMAP working papers 23/12, Institute for Fiscal Studies.
    2. Michael Vogt & Oliver Linton, 2014. "Nonparametric estimation of a periodic sequence in the presence of a smooth trend," Biometrika, Biometrika Trust, vol. 101(1), pages 121-140.
    3. Zheng Xu, 2016. "An alternative circular smoothing method to nonparametric estimation of periodic functions," Journal of Applied Statistics, Taylor & Francis Journals, vol. 43(9), pages 1649-1672, July.
    4. Arthur Pewsey & Eduardo García-Portugués, 2021. "Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 1-58, March.

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