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Title: Bootstrap confidence bands for spectral estimation of L\'{e}vy densities under high-frequency observations Authors:  Daisuke Kurisu - Yokohama National University (Japan) [presenting]
Kengo Kato - Cornell University (United States)
Abstract: Bootstrap methods are developed to construct uniform confidence bands for nonparametric spectral estimation of L\'{e}vy 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 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, conduct simulation studies to investigate the finite sample performance of the proposed confidence bands, and present a small real data analysis.