View source for Spectral density, estimator of the

<!--
s0863801.png
$#A+1 = 34 n = 0
$#C+1 = 34 : ~/encyclopedia/old_files/data/S086/S.0806380 Spectral density, estimator of the
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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

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

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|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|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|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) {{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====

====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) {{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>