Spectral density, estimator of the

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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

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


is a sample estimator of the covariance function of the stationary process and the are suitably chosen weights. The estimator can be written as

where is the periodogram and is some continuous even function with of its Fourier coefficients specified:

The function is called a spectral window; one usually considers spectral windows of the form

where is some continuous function on such that

and as , but . Similarly, one considers coefficients of the form

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).


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How to Cite This Entry:
Spectral density, estimator of the. Encyclopedia of Mathematics. URL:,_estimator_of_the&oldid=24129
This article was adapted from an original article by I.G. Zhurbenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article