From Adrian.Trapletti@wu-wien.ac.at  Tue Jul 20 13:53:28 1999
From: Adrian.Trapletti@wu-wien.ac.at (Adrian Trapletti)
Date: Tue, 20 Jul 1999 12:53:28 +0000
Subject: time series in R
Message-ID: <37947148.EB23277B@wu-wien.ac.at>

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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time series in R
Adrian Trapletti