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Adaptive pointwise estimation for pure jump Lévy processes

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  • Mélina Bec
  • Claire Lacour

Abstract

This paper is concerned with adaptive kernel estimation of the Lévy density $$N(x)$$ N ( x ) for bounded-variation pure-jump Lévy processes. The sample path is observed at $$n$$ n discrete instants in the “high frequency” context ( $$\Delta=\Delta (n) $$ Δ = Δ ( n ) tends to zero while $$n \Delta $$ n Δ tends to infinity). We construct a collection of kernel estimators of the function $$g(x)=xN(x)$$ g ( x ) = x N ( x ) and propose a method of local adaptive selection of the bandwidth. We provide an oracle inequality and a rate of convergence for the quadratic pointwise risk. This rate is proved to be the optimal minimax rate. We give examples and simulation results for processes fitting in our framework. We also consider the case of irregular sampling. Copyright Springer Science+Business Media Dordrecht 2015

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  • Mélina Bec & Claire Lacour, 2015. "Adaptive pointwise estimation for pure jump Lévy processes," Statistical Inference for Stochastic Processes, Springer, vol. 18(3), pages 229-256, October.
    • Handle: RePEc:spr:sistpr:v:18:y:2015:i:3:p:229-256
    DOI: 10.1007/s11203-014-9113-6
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    Cited by:

    1. Kato, Kengo & Kurisu, Daisuke, 2020. "Bootstrap confidence bands for spectral estimation of Lévy densities under high-frequency observations," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1159-1205.

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