Spectral density, estimator of the
A function of the observed values of a discrete-time stationary stochastic process, used as an estimator of the spectral density
. As an estimator of the spectral density one often uses quadratic forms
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where the are complex coefficients (depending on
). It can be shown that the asymptotic behaviour as
of the first two moments of an estimator of the spectral density is satisfactory, in general, if one considers only the subclass of quadratic forms such that
when
. This enables one to restrict attention to estimators of the spectral density of the form
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where
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is a sample estimator of the covariance function of the stationary process and the
are suitably chosen weights. The estimator
can be written as
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where is the periodogram and
is some continuous even function with
of its Fourier coefficients specified:
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The function is called a spectral window; one usually considers spectral windows of the form
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where is some continuous function on
such that
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and as
, but
. Similarly, one considers coefficients
of the form
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and a function , called a lag window or covariance window. Under weak smoothness restrictions on the spectral density
, or assuming that
is mixing, it is possible to prove that for a wide class of spectral or covariance windows the estimator
is asymptotically unbiased and consistent.
In the case of a multi-dimensional stochastic process, estimation of the elements of the matrix of spectral densities proceeds in a similar way using the corresponding periodogram
. Instead of an estimator of the spectral density in the form of a quadratic form in the observations, one often assumes that the spectral density depends in a particular way on a finite number of parameters, and then one seeks estimators based on the observations of the parameters involved in this expression for the spectral density (see Maximum-entropy spectral estimator; Spectral estimator, parametric).
References
[1] | D.R. Brillinger, "Time series. Data analysis and theory" , Holt, Rinehart & Winston (1975) |
[2] | E.J. Hannan, "Multiple time series" , Wiley (1972) |
[3] | T.M. Anderson, "Statistical analysis of time series" , Wiley (1971) |
Comments
References
[a1] | G.E.P. Box, G.M. Jenkins, "Time series analysis. Forecasting and control" , Holden-Day (1960) |
[a2] | P.E. Caines, "Linear stochastic systems" , Wiley (1988) |
[a3] | K.O. Dzhaparidze, "Parameter estimation and hypothesis testing in spectral analysis of stationary time series" , Springer (1986) |
[a4] | L. Ljung, "System identification theory for the user" , Prentice-Hall (1987) |
Spectral density, estimator of the. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_density,_estimator_of_the&oldid=13745