View source for Spectral density, estimator of the

A function of the observed values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s0863801.png" /> of a discrete-time [[Stationary stochastic process|stationary stochastic process]], used as an estimator of the [[Spectral density|spectral density]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s0863802.png" />. As an estimator of the spectral density one often uses quadratic forms

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s0863803.png" /></td> </tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s0863804.png" /> are complex coefficients (depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s0863805.png" />). It can be shown that the asymptotic behaviour as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s0863806.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s0863807.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s0863808.png" />. This enables one to restrict attention to estimators of the spectral density of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s0863809.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638010.png" /></td> </tr></table>

is a sample estimator of the covariance function of the stationary process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638011.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638012.png" /> are suitably chosen weights. The estimator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638013.png" /> can be written as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638014.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638015.png" /> is the [[Periodogram|periodogram]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638016.png" /> is some continuous even function with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638017.png" /> of its Fourier coefficients specified:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638018.png" /></td> </tr></table>

The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638019.png" /> is called a [[Spectral window|spectral window]]; one usually considers spectral windows of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638020.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638021.png" /> is some continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638022.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638023.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638024.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638025.png" />, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638026.png" />. Similarly, one considers coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638027.png" /> of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638028.png" /></td> </tr></table>

and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638029.png" />, called a lag window or covariance window. Under weak smoothness restrictions on the spectral density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638030.png" />, or assuming that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638031.png" /> is mixing, it is possible to prove that for a wide class of spectral or covariance windows the estimator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638032.png" /> is asymptotically unbiased and consistent.

In the case of a multi-dimensional stochastic process, estimation of the elements of the matrix of spectral densities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638033.png" /> proceeds in a similar way using the corresponding periodogram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086380/s08638034.png" />. 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|Maximum-entropy spectral estimator]]; [[Spectral estimator, parametric|Spectral estimator, parametric]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.R. Brillinger,   "Time series. Data analysis and theory" , Holt, Rinehart &amp; Winston  (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.J. Hannan,   "Multiple time series" , Wiley  (1972)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  T.M. Anderson,   "Statistical analysis of time series" , Wiley  (1971)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.E.P. Box,   G.M. Jenkins,   "Time series analysis. Forecasting and control" , Holden-Day  (1960)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.E. Caines,   "Linear stochastic systems" , Wiley  (1988)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K.O. Dzhaparidze,   "Parameter estimation and hypothesis testing in spectral analysis of stationary time series" , Springer  (1986)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Ljung,   "System identification theory for the user" , Prentice-Hall  (1987)</TD></TR></table>