View source for Spectral function, estimator of the

''estimator of the spectral measure''

A function of the observed values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086410/s0864101.png" /> of a discrete-time [[Stationary stochastic process|stationary stochastic process]], used as an estimator of the [[Spectral function|spectral function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086410/s0864102.png" />. As an estimator of this function one often uses an expression 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/s086410/s0864103.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086410/s0864104.png" /> is the [[Periodogram|periodogram]]. Under fairly general smoothness conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086410/s0864105.png" />, or under mixing conditions on the random process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086410/s0864106.png" />, this estimator turns out to be asymptotically unbiased and consistent.

The above estimator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086410/s0864107.png" /> is a special case of an estimator

<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/s086410/s0864108.png" /></td> </tr></table>

of a function

<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/s086410/s0864109.png" /></td> </tr></table>

of the spectral density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086410/s08641010.png" />. In particular, many estimators of the spectral density (cf. [[Spectral density, estimator of the|Spectral density, estimator of the]]) reduce to this form, where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086410/s08641011.png" /> depends on the size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086410/s08641012.png" /> of the sample and is concentrated about the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086410/s08641013.png" />.

====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  (1970)</TD></TR></table>



====Comments====


====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></table>