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The spectrum of a stochastic process occurring in the regression scheme for a stationary time series. Thus, let a [[Stochastic process|stochastic process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r0806501.png" /> which is observable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r0806502.png" /> be represented in the form
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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/r/r080/r080650/r0806503.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r0806504.png" /> is a [[Stationary stochastic process|stationary stochastic process]] with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r0806505.png" />, and let the mean value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r0806506.png" /> be expressed in the form of a linear [[Regression|regression]]
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The spectrum of a stochastic process occurring in the regression scheme for a stationary time series. Thus, let a [[Stochastic process|stochastic process]] $  y _ {t} $
 +
which is observable for  $  t = 1 \dots n $
 +
be represented in 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/r/r080/r080650/r0806507.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \tag{1 }
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y _ {t}  = m _ {t} + x _ {t} ,
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r0806508.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r0806509.png" />, are known regression vectors and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065010.png" /> are unknown regression coefficients (cf. [[Regression coefficient|Regression coefficient]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065011.png" /> be the spectral distribution function of the regression vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065012.png" /> (cf. [[Spectral analysis of a stationary stochastic process|Spectral analysis of a stationary stochastic process]]). The regression spectrum for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065013.png" /> is the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065014.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065015.png" /> for any interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065016.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065018.png" />.
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where $  x _ {t} $
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is a [[Stationary stochastic process|stationary stochastic process]] with  $  {\mathsf E} x _ {t} \equiv 0 $,
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and let the mean value  $  {\mathsf E} y _ {t} = m _ {t} $
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be expressed in the form of a linear [[Regression|regression]]
  
The regression spectrum plays an important role in problems of estimating the regression coefficients in the scheme (1)–(2). For example, the elements of a regression spectrum can be used to express a necessary and sufficient condition for the asymptotic efficiency of an estimator for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080650/r08065019.png" /> by the method of least squares.
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$$ \tag{2 }
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m _ {t}  = \
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\sum _ { k= } 1 ^ { s }
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\beta _ {k} \phi _ {t}  ^ {(} k) ,
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$$
 +
 
 +
where  $  \phi  ^ {(} k) = ( \phi _ {1}  ^ {(} k) \dots \phi _ {n}  ^ {(} k) ) $,
 +
$  k = 1 \dots s $,
 +
are known regression vectors and  $  \beta _ {1} \dots \beta _ {s} $
 +
are unknown regression coefficients (cf. [[Regression coefficient|Regression coefficient]]). Let  $  M ( \lambda ) $
 +
be the spectral distribution function of the regression vectors  $  \phi  ^ {(} 1) \dots \phi  ^ {(} s) $(
 +
cf. [[Spectral analysis of a stationary stochastic process|Spectral analysis of a stationary stochastic process]]). The regression spectrum for  $  M ( \lambda ) $
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is the set of all  $  \lambda $
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such that  $  M ( \lambda _ {2} ) - M ( \lambda _ {1} ) > 0 $
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for any interval  $  ( \lambda _ {1} , \lambda _ {2} ) $
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containing  $  \lambda $,
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$  \lambda _ {1} < \lambda < \lambda _ {2} $.
 +
 
 +
The regression spectrum plays an important role in problems of estimating the regression coefficients in the scheme (1)–(2). For example, the elements of a regression spectrum can be used to express a necessary and sufficient condition for the asymptotic efficiency of an estimator for $  \beta $
 +
by the method of least squares.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  U. Grenander,  M. Rosenblatt,  "Statistical analysis of stationary time series" , Wiley  (1957)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  U. Grenander,  M. Rosenblatt,  "Statistical analysis of stationary time series" , Wiley  (1957)</TD></TR></table>

Revision as of 08:10, 6 June 2020


The spectrum of a stochastic process occurring in the regression scheme for a stationary time series. Thus, let a stochastic process $ y _ {t} $ which is observable for $ t = 1 \dots n $ be represented in the form

$$ \tag{1 } y _ {t} = m _ {t} + x _ {t} , $$

where $ x _ {t} $ is a stationary stochastic process with $ {\mathsf E} x _ {t} \equiv 0 $, and let the mean value $ {\mathsf E} y _ {t} = m _ {t} $ be expressed in the form of a linear regression

$$ \tag{2 } m _ {t} = \ \sum _ { k= } 1 ^ { s } \beta _ {k} \phi _ {t} ^ {(} k) , $$

where $ \phi ^ {(} k) = ( \phi _ {1} ^ {(} k) \dots \phi _ {n} ^ {(} k) ) $, $ k = 1 \dots s $, are known regression vectors and $ \beta _ {1} \dots \beta _ {s} $ are unknown regression coefficients (cf. Regression coefficient). Let $ M ( \lambda ) $ be the spectral distribution function of the regression vectors $ \phi ^ {(} 1) \dots \phi ^ {(} s) $( cf. Spectral analysis of a stationary stochastic process). The regression spectrum for $ M ( \lambda ) $ is the set of all $ \lambda $ such that $ M ( \lambda _ {2} ) - M ( \lambda _ {1} ) > 0 $ for any interval $ ( \lambda _ {1} , \lambda _ {2} ) $ containing $ \lambda $, $ \lambda _ {1} < \lambda < \lambda _ {2} $.

The regression spectrum plays an important role in problems of estimating the regression coefficients in the scheme (1)–(2). For example, the elements of a regression spectrum can be used to express a necessary and sufficient condition for the asymptotic efficiency of an estimator for $ \beta $ by the method of least squares.

References

[1] U. Grenander, M. Rosenblatt, "Statistical analysis of stationary time series" , Wiley (1957)
How to Cite This Entry:
Regression spectrum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regression_spectrum&oldid=48476
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article