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Spectral function, estimator of the

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estimator of the spectral measure

A function of the observed values of a discrete-time stationary stochastic process, used as an estimator of the spectral function . As an estimator of this function one often uses an expression of the form

where is the periodogram. Under fairly general smoothness conditions on , or under mixing conditions on the random process , this estimator turns out to be asymptotically unbiased and consistent.

The above estimator of is a special case of an estimator

of a function

of the spectral density . In particular, many estimators of the spectral density (cf. Spectral density, estimator of the) reduce to this form, where the function depends on the size of the sample and is concentrated about the point .

References

[1] D.R. Brillinger, "Time series. Data analysis and theory" , Holt, Rinehart & Winston (1975)
[2] E.J. Hannan, "Multiple time series" , Wiley (1970)


Comments

References

[a1] G.E.P. Box, G.M. Jenkins, "Time series analysis. Forecasting and control" , Holden-Day (1960)
How to Cite This Entry:
Spectral function, estimator of the. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_function,_estimator_of_the&oldid=33166
This article was adapted from an original article by I.G. Zhurbenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article