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''estimator of the spectral measure''
 
''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
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A function of the observed values $X(1),\dots,X(N)$ of a discrete-time [[Stationary stochastic process|stationary stochastic process]], used as an estimator of the [[Spectral function|spectral function]] $F(\lambda)$. 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>
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$$F_N(\lambda)=\frac{2\pi}{N}\sum_{-\pi\leq2\pi k/N\leq\lambda}I_N\left(\frac{2\pi k}{N}\right),$$
  
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.
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where $I_N(x)$ is the [[Periodogram|periodogram]]. Under fairly general smoothness conditions on $F(\lambda)$, or under mixing conditions on the random process $X(t)$, 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
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The above estimator of $F(\lambda)$ 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>
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$$\frac{2\pi}{N}\sum_{-\pi\leq2\pi k/N\leq\pi}A\left(\frac{2\pi k}{N}\right)I_N\left(\frac{2\pi k}{N}\right)$$
  
 
of a function
 
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>
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$$I(A)=\int\limits_{-\pi}^\pi A(x)f(x)dx$$
  
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" />.
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of the spectral density $f(\lambda)$. 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 $A(x)$ depends on the size $N$ of the sample and is concentrated about the point $x=\lambda$.
  
 
====References====
 
====References====

Latest revision as of 13:47, 27 August 2014

estimator of the spectral measure

A function of the observed values $X(1),\dots,X(N)$ of a discrete-time stationary stochastic process, used as an estimator of the spectral function $F(\lambda)$. As an estimator of this function one often uses an expression of the form

$$F_N(\lambda)=\frac{2\pi}{N}\sum_{-\pi\leq2\pi k/N\leq\lambda}I_N\left(\frac{2\pi k}{N}\right),$$

where $I_N(x)$ is the periodogram. Under fairly general smoothness conditions on $F(\lambda)$, or under mixing conditions on the random process $X(t)$, this estimator turns out to be asymptotically unbiased and consistent.

The above estimator of $F(\lambda)$ is a special case of an estimator

$$\frac{2\pi}{N}\sum_{-\pi\leq2\pi k/N\leq\pi}A\left(\frac{2\pi k}{N}\right)I_N\left(\frac{2\pi k}{N}\right)$$

of a function

$$I(A)=\int\limits_{-\pi}^\pi A(x)f(x)dx$$

of the spectral density $f(\lambda)$. In particular, many estimators of the spectral density (cf. Spectral density, estimator of the) reduce to this form, where the function $A(x)$ depends on the size $N$ of the sample and is concentrated about the point $x=\lambda$.

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