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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
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<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>
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{{TEX|auto}}
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{{TEX|done}}
  
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
+
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 density|spectral density]]  $  f ( \lambda ) $.
 +
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/s0863809.png" /></td> </tr></table>
+
$$
 +
\sum _ {s , t = 1 } ^ { N }
 +
b _ {s,t}  ^ {(N)} X ( s) X ( t) ,
 +
$$
 +
 
 +
where the  $  b _ {s,t}  ^ {(N)} $
 +
are complex coefficients (depending on  $  \lambda $).
 +
It can be shown that the asymptotic behaviour as  $  N \rightarrow \infty $
 +
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  $  b _ {s _ {1}  , t _ {1} }  ^ {(N)} = b _ {s _ {2}  , t _ {2} }  ^ {(N)} $
 +
when  $  s _ {1} - t _ {1} = s _ {2} - t _ {2} $.
 +
This enables one to restrict attention to estimators of the spectral density of the form
 +
 
 +
$$
 +
\widehat{f}  _ {N} ( \lambda )  =
 +
\frac{1}{2 \pi } \sum _ {t = - N + 1 } ^ {N-1} e ^ {i t \lambda } b _ {N} ( t) B _ {N} ( t) ,
 +
$$
  
 
where
 
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>
+
$$
 +
B _ {N} ( t)  =
 +
\frac{1}{N} \sum _{s=1}^ { {N }  - | t | }
 +
X ( s) X ( s + | t | )
 +
$$
  
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
+
is a sample estimator of the covariance function of the stationary process $  X ( t) $
 +
and the $  b _ {N} ( t) $
 +
are suitably chosen weights. The estimator $  \widehat{f}  _ {N} ( \lambda ) $
 +
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>
+
$$
 +
\widehat{f}  _ {N} ( \lambda )  = \
 +
\int\limits _ {- \pi } ^  \pi 
 +
\Phi _ {N} ( x) I _ {N} ( x + \lambda )  d x ,
 +
$$
  
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:
+
where $  I _ {N} ( x) $
 +
is the [[Periodogram|periodogram]] and $  \Phi _ {N} ( x) $
 +
is some continuous even function with $  2N- 1 $
 +
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>
+
$$
 +
b _ {N} ( t)  = \int\limits _ {- \pi } ^  \pi 
 +
\Phi _ {N} ( x) e ^ {i t x }  d x ,\ \
 +
t = - N + 1 \dots N - 1 .
 +
$$
  
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
+
The function $  \Phi _ {N} ( x) $
 +
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>
+
$$
 +
\Phi _ {N} ( x)  = A _ {N} \Phi ( A _ {N} x ) ,
 +
$$
  
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
+
where $  \Phi ( x) $
 +
is some continuous function on $  ( - \infty , \infty ) $
 +
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>
+
$$
 +
\int\limits _ {- \infty } ^  \infty  \Phi ( x)  d x  = 1 ,
 +
$$
  
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
+
and $  A _ {N} \rightarrow \infty $
 +
as $  N \rightarrow \infty $,  
 +
but $  A _ {N} N  ^ {-1} \rightarrow 0 $.  
 +
Similarly, one considers coefficients $  b _ {N} ( t) $
 +
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>
+
$$
 +
b _ {N} ( t)  = K ( A _ {N}  ^ {-1} t )
 +
$$
  
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.
+
and a function $  K ( x) $,  
 +
called a lag window or covariance window. Under weak smoothness restrictions on the spectral density $  f ( \lambda ) $,  
 +
or assuming that $  X ( t) $
 +
is mixing, it is possible to prove that for a wide class of spectral or covariance windows the estimator $  \widehat{f}  _ {N} ( \lambda ) $
 +
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]]).
+
In the case of a multi-dimensional stochastic process, estimation of the elements of the matrix of spectral densities $  f _ {k,l} ( \lambda ) $
 +
proceeds in a similar way using the corresponding periodogram $  I _ {N} ^ {( k , l ) } ( \lambda ) $.  
 +
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====
 
====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>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D.R. Brillinger, "Time series. Data analysis and theory" , Holt, Rinehart &amp; Winston (1975) {{MR|0443257}} {{ZBL|0321.62004}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E.J. Hannan, "Multiple time series" , Wiley (1972) {{MR|0279952}} {{ZBL|0279.62025}} {{ZBL|0211.49804}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> T.M. Anderson, "Statistical analysis of time series" , Wiley (1971) {{MR|0283939}} {{ZBL|0225.62108}} </TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====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>
+
<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) {{MR|0436499}} {{MR|0353595}} {{MR|0353594}} {{MR|0272138}} {{ZBL|1154.62062}} {{ZBL|0858.62072}} {{ZBL|0363.62069}} {{ZBL|0284.62059}} {{ZBL|0276.62080}} {{ZBL|0249.62009}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.E. Caines, "Linear stochastic systems" , Wiley (1988) {{MR|0944080}} {{ZBL|0658.93003}} </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) {{MR|0775857}} {{MR|0812272}} {{ZBL|0584.62157}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> L. Ljung, "System identification theory for the user" , Prentice-Hall (1987) {{MR|1157156}} {{ZBL|0615.93004}} </TD></TR></table>

Latest revision as of 19:58, 16 January 2024


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 density $ f ( \lambda ) $. As an estimator of the spectral density one often uses quadratic forms

$$ \sum _ {s , t = 1 } ^ { N } b _ {s,t} ^ {(N)} X ( s) X ( t) , $$

where the $ b _ {s,t} ^ {(N)} $ are complex coefficients (depending on $ \lambda $). It can be shown that the asymptotic behaviour as $ N \rightarrow \infty $ 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 $ b _ {s _ {1} , t _ {1} } ^ {(N)} = b _ {s _ {2} , t _ {2} } ^ {(N)} $ when $ s _ {1} - t _ {1} = s _ {2} - t _ {2} $. This enables one to restrict attention to estimators of the spectral density of the form

$$ \widehat{f} _ {N} ( \lambda ) = \frac{1}{2 \pi } \sum _ {t = - N + 1 } ^ {N-1} e ^ {i t \lambda } b _ {N} ( t) B _ {N} ( t) , $$

where

$$ B _ {N} ( t) = \frac{1}{N} \sum _{s=1}^ { {N } - | t | } X ( s) X ( s + | t | ) $$

is a sample estimator of the covariance function of the stationary process $ X ( t) $ and the $ b _ {N} ( t) $ are suitably chosen weights. The estimator $ \widehat{f} _ {N} ( \lambda ) $ can be written as

$$ \widehat{f} _ {N} ( \lambda ) = \ \int\limits _ {- \pi } ^ \pi \Phi _ {N} ( x) I _ {N} ( x + \lambda ) d x , $$

where $ I _ {N} ( x) $ is the periodogram and $ \Phi _ {N} ( x) $ is some continuous even function with $ 2N- 1 $ of its Fourier coefficients specified:

$$ b _ {N} ( t) = \int\limits _ {- \pi } ^ \pi \Phi _ {N} ( x) e ^ {i t x } d x ,\ \ t = - N + 1 \dots N - 1 . $$

The function $ \Phi _ {N} ( x) $ is called a spectral window; one usually considers spectral windows of the form

$$ \Phi _ {N} ( x) = A _ {N} \Phi ( A _ {N} x ) , $$

where $ \Phi ( x) $ is some continuous function on $ ( - \infty , \infty ) $ such that

$$ \int\limits _ {- \infty } ^ \infty \Phi ( x) d x = 1 , $$

and $ A _ {N} \rightarrow \infty $ as $ N \rightarrow \infty $, but $ A _ {N} N ^ {-1} \rightarrow 0 $. Similarly, one considers coefficients $ b _ {N} ( t) $ of the form

$$ b _ {N} ( t) = K ( A _ {N} ^ {-1} t ) $$

and a function $ K ( x) $, called a lag window or covariance window. Under weak smoothness restrictions on the spectral density $ f ( \lambda ) $, or assuming that $ X ( t) $ is mixing, it is possible to prove that for a wide class of spectral or covariance windows the estimator $ \widehat{f} _ {N} ( \lambda ) $ is asymptotically unbiased and consistent.

In the case of a multi-dimensional stochastic process, estimation of the elements of the matrix of spectral densities $ f _ {k,l} ( \lambda ) $ proceeds in a similar way using the corresponding periodogram $ I _ {N} ^ {( k , l ) } ( \lambda ) $. 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).

References

[1] D.R. Brillinger, "Time series. Data analysis and theory" , Holt, Rinehart & Winston (1975) MR0443257 Zbl 0321.62004
[2] E.J. Hannan, "Multiple time series" , Wiley (1972) MR0279952 Zbl 0279.62025 Zbl 0211.49804
[3] T.M. Anderson, "Statistical analysis of time series" , Wiley (1971) MR0283939 Zbl 0225.62108

Comments

References

[a1] G.E.P. Box, G.M. Jenkins, "Time series analysis. Forecasting and control" , Holden-Day (1960) MR0436499 MR0353595 MR0353594 MR0272138 Zbl 1154.62062 Zbl 0858.62072 Zbl 0363.62069 Zbl 0284.62059 Zbl 0276.62080 Zbl 0249.62009
[a2] P.E. Caines, "Linear stochastic systems" , Wiley (1988) MR0944080 Zbl 0658.93003
[a3] K.O. Dzhaparidze, "Parameter estimation and hypothesis testing in spectral analysis of stationary time series" , Springer (1986) MR0775857 MR0812272 Zbl 0584.62157
[a4] L. Ljung, "System identification theory for the user" , Prentice-Hall (1987) MR1157156 Zbl 0615.93004
How to Cite This Entry:
Spectral density, estimator of the. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_density,_estimator_of_the&oldid=13745
This article was adapted from an original article by I.G. Zhurbenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article