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Bootstrap confidence bands for spectral estimation of Lévy densities under high-frequency observations

1 May 2017
Kengo Kato
Daisuke Kurisu
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Abstract

This paper develops bootstrap methods to construct uniform confidence bands for nonparametric spectral estimation of L\'{e}vy densities under high-frequency observations. We assume that we observe nnn discrete observations at frequency 1/Δ>01/\Delta > 01/Δ>0, and work with the high-frequency setup where Δ=Δn→0\Delta = \Delta_{n} \to 0Δ=Δn​→0 and nΔ→∞n\Delta \to \inftynΔ→∞ as n→∞n \to \inftyn→∞. We employ a spectral (or Fourier-based) estimator of the L\'{e}vy density, and develop novel implementations of Gaussian multiplier (or wild) and empirical (or Efron's) bootstraps to construct confidence bands for the spectral estimator on a compact set that does not intersect the origin. We provide conditions under which the proposed confidence bands are asymptotically valid. Our confidence bands are shown to be asymptotically valid for a wide class of L\'{e}vy processes. We also develop a practical method for bandwidth selection, and conduct simulation studies to investigate the finite sample performance of the proposed confidence bands.

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