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A solution to a linear integral equation with an application to statistics of infinitely divisible moving averages

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  • Jochen Glück
  • Stefan Roth
  • Evgeny Spodarev

Abstract

For a stationary moving average random field, a nonparametric low frequency estimator of the Lévy density of its infinitely divisible independently scattered integrator measure is given. The plug‐in estimate is based on the solution w of the linear integral equation v(x)=∫ℝdg(s)w(h(s)x)ds, where g,h:ℝd→ℝ are given measurable functions and v is a (weighted) L2‐function on ℝ. We investigate conditions for the existence and uniqueness of this solution and give L2‐error bounds for the resulting estimates. An application to pure jump moving averages and a simulation study round off the paper.

Suggested Citation

  • Jochen Glück & Stefan Roth & Evgeny Spodarev, 2022. "A solution to a linear integral equation with an application to statistics of infinitely divisible moving averages," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(3), pages 1244-1273, September.
    • Handle: RePEc:bla:scjsta:v:49:y:2022:i:3:p:1244-1273
    DOI: 10.1111/sjos.12553
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    References listed on IDEAS

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    1. Trabs, Mathias, 2014. "On infinitely divisible distributions with polynomially decaying characteristic functions," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 56-62.
    2. Wolfgang Karcher & Stefan Roth & Evgeny Spodarev & Corinna Walk, 2019. "An inverse problem for infinitely divisible moving average random fields," Statistical Inference for Stochastic Processes, Springer, vol. 22(2), pages 263-306, July.
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    1. Trabs, Mathias, 2015. "Quantile estimation for Lévy measures," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3484-3521.
    2. Wolfgang Karcher & Stefan Roth & Evgeny Spodarev & Corinna Walk, 2019. "An inverse problem for infinitely divisible moving average random fields," Statistical Inference for Stochastic Processes, Springer, vol. 22(2), pages 263-306, July.

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