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Biometrika Advance Access originally published online on April 24, 2008
Biometrika 2008 95(2):381-397; doi:10.1093/biomet/asn005
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© 2008 Biometrika Trust

Articles

Simultaneous confidence bands in spectral density estimation

Michael H. Neumann

Institut für Stochastik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, D–07743 Jena, Germany mneumann{at}mathematik.uni-jena.de

Efstathios Paparoditis

Department of Mathematics, and Statistics, University of Cyprus, P.O. Box 20537, CY–1678 Nicosia, Cyprus stathisp{at}ucy.ac.cy

Received for publication 1 July 2006. Revision received 1 September 2007.
  Abstract

We propose a method for the construction of simultaneous confidence bands for a smoothed version of the spectral density of a Gaussian process based on nonparametric kernel estimators obtained by smoothing the periodogram. A studentized statistic is used to determine the width of the band at each frequency and a frequency-domain bootstrap approach is employed to estimate the distribution of the supremum of this statistic over all frequencies. We prove by means of strong approximations that the bootstrap estimates consistently the distribution of the supremum deviation of interest and, consequently, that the proposed confidence bands achieve asymptotically the desired simultaneous coverage probability. The behaviour of our method in finite-sample situations is investigated by simulations and a real-life data example demonstrates its applicability in time series analysis.

Key Words: Bootstrap • Confidence band • Gaussian process • Spectral density • Strong approximation


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