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Biometrika 2004 91(1):211-218; doi:10.1093/biomet/91.1.211
© 2004 by Biometrika Trust
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Contiguity of the Whittle measure for a Gaussian time series

Nidhan Choudhuri1, Subhashis Ghosal2 and Anindya Roy3

1 Department of Statistics, Case Western Reserve University, Cleveland, Ohio 44106, U.S.Anidhan{at}nidhan.cwru.edu 2 Department of Statistics, North Carolina State University, Raleigh, North Carolina 27695, U.S.A.ghosal{at}stat.ncsu.edu 3 Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, Maryland 21250, U.S.A. anindya{at}math.umbc.edu

For a stationary time series, Whittle constructed a likelihood for the spectral density based on the approximate independence of the discrete Fourier transforms of the data at certain frequencies. Whittle's likelihood has been widely used in the literature for constructing estimators.In this paper, we show that, for a Gaussian time series, the Whittle measure is mutually contiguous with the actual distribution of the data. As a consequence, most asymptotic properties of estimators and test statistics derived under the Whittle measure can be carried over to the actual distribution.

Key Words: Consistency; Contiguity; Discrete Fourier transform; Periodogram; Spectral density; Whittle likelihood

Received October 2002. Revised July 2003


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