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An inverse problem for infinitely divisible moving average random fields

Author

Listed:
  • Wolfgang Karcher

    (Ulm University)

  • Stefan Roth

    (Ulm University)

  • Evgeny Spodarev

    (Ulm University)

  • Corinna Walk

    (Ulm University)

Abstract

Given a low frequency sample of an infinitely divisible moving average random field $$\{\int _{\mathbb {R}^d} f(x-t)\varLambda (dx); \ t \in \mathbb {R}^d \}$$ { ∫ R d f ( x - t ) Λ ( d x ) ; t ∈ R d } with a known simple function f, we study the problem of nonparametric estimation of the Lévy characteristics of the independently scattered random measure $$\varLambda $$ Λ . We provide three methods, a simple plug-in approach, a method based on Fourier transforms and an approach involving decompositions with respect to $$L^2$$ L 2 -orthonormal bases, which allow to estimate the Lévy density of $$\varLambda $$ Λ . For these methods, the bounds for the $$L^2$$ L 2 -error are given. Their numerical performance is compared in a simulation study.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:sistpr:v:22:y:2019:i:2:d:10.1007_s11203-018-9188-6
    DOI: 10.1007/s11203-018-9188-6
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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. Shota Gugushvili & Frank Meulen & Peter Spreij, 2018. "A non-parametric Bayesian approach to decompounding from high frequency data," Statistical Inference for Stochastic Processes, Springer, vol. 21(1), pages 53-79, April.
    3. Comte, F. & Genon-Catalot, V., 2009. "Nonparametric estimation for pure jump Lévy processes based on high frequency data," Stochastic Processes and their Applications, Elsevier, vol. 119(12), pages 4088-4123, December.
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    Cited by:

    1. 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.

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