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

摘要

This paper develops bootstrap methods to construct uniform confidence bandsfor nonparametric spectral estimation of L'{e}vy densities underhigh-frequency observations. We assume that we observe $n$ discreteobservations at frequency $1/Delta > 0$, and work with the high-frequencysetup where $Delta = Delta_{n} o 0$ and $nDelta o infty$ as $n oinfty$. We employ a spectral (or Fourier-based) estimator of the L'{e}vydensity, and develop novel implementations of Gaussian multiplier (or wild) andempirical (or Efron's) bootstraps to construct confidence bands for thespectral estimator on a compact set that does not intersect the origin. Weprovide conditions under which the proposed confidence bands are asymptoticallyvalid. Our confidence bands are shown to be asymptotically valid for a wideclass of L'{e}vy processes. We also develop a practical method for bandwidthselection, and conduct simulation studies to investigate the finite sampleperformance of the proposed confidence bands.