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

Abstract

This paper develops bootstrap methods to construct uniform confidence bands for nonparametric spectral estimation of Lévy densities under high-frequency observations. We assume that we observe $n$ discrete observations at frequency $1/\Delta >0$, and work with the high-frequency setup where $\Delta ={\Delta }_n\to 0$ and $n\Delta \to \infty$ as $n\to \infty$. We employ a spectral (or Fourier-based) estimator of the Lévy 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 Lévy 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.

50pages