# time series in R

**Adrian Trapletti
**
Adrian.Trapletti@wu-wien.ac.at

*Tue, 20 Jul 1999 12:53:28 +0000*

Martin Maechler wrote:
>*
*>* Something which should be discussed however is spectrum(0);
*>* Several of us think that S-plus does the wrong thing, at least in some
*
>* cases. If demean=T (mean removed), should have periodogram(0) = 0,
*>* and maybe even spectrum(0) = 0 [and hence dB-spec. = -Inf ..]
*>* Another possibility would be to leave it NA
*>* and maybe provide methods for estimating it specifically, if desired.
*>*
*
I had a look at some of our Dep. books:
Brockwell&Davis: Periodogram normalization is n^{-1}, P(0)=0 for
demean=T.
spectrum(0) should be estimated by not using P(0) (Remark 2, p. 353). In
general
S(0) \neq 0.
Shiryaev, Probability: Per. norm. is (2*pi*n)^{-1}, P(0)=0 for demean=T.
Priestley, Spectral Analysis... : Periodogram normalization is
(n/2)^{-1}, P(0)=0
for demean=T, p. 395. For continuous spectra he defines a "modified
Periodogram",pp. 416, 417, where the normalization is as in Shiryaev.
All the
spectrum estimation is done with the mod. Period.
Hannan, Multiple Time Series: Normalization is (n/2)^{-1}.
Koopmans: Spectral Analysis of TS: Norm. is (2*pi*n)^{-1}.
It seems that (2*pi*n)^{-1} is the version which is mostly used, since
it makes no
further normalization necessary, e.g., for smoothing the periodogram.
P(0)=0 is
obvious. And \hat{spectrum}(0) = 0 is definitely a very bad estimator.
Adrian
--
Adrian Trapletti, Vienna University of Economics and Business
Administration, Augasse 2-6, A-1090 Vienna, Austria
Phone: ++43 1 31336 4561, Fax: ++43 1 31336 708,
Email: adrian.trapletti@wu-wien.ac.at
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