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Difference between revisions of "Regression spectrum"

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$$ \tag{2 }
 
$$ \tag{2 }
m _ {t}  = \  
+
m _ {t}  = \sum_{k=1}^ { s } \beta _ {k} \phi _ {t}  ^ {(k)} ,
\sum _ { k= } 1 ^ { s }  
 
\beta _ {k} \phi _ {t}  ^ {(} k) ,
 
 
$$
 
$$
  
where  $  \phi  ^ {(} k) = ( \phi _ {1}  ^ {(} k) \dots \phi _ {n}  ^ {(} k) ) $,  
+
where  $  \phi  ^ {(k)} = ( \phi _ {1}  ^ {(k)} \dots \phi _ {n}  ^ {(k)} ) $,  
 
$  k = 1 \dots s $,  
 
$  k = 1 \dots s $,  
 
are known regression vectors and  $  \beta _ {1} \dots \beta _ {s} $
 
are known regression vectors and  $  \beta _ {1} \dots \beta _ {s} $
are unknown regression coefficients (cf. [[Regression coefficient|Regression coefficient]]). Let  $  M ( \lambda ) $
+
are unknown regression coefficients (cf. [[Regression coefficient]]). Let  $  M ( \lambda ) $
be the spectral distribution function of the regression vectors  $  \phi  ^ {(} 1) \dots \phi  ^ {(} s) $(
+
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 ) $
 
cf. [[Spectral analysis of a stationary stochastic process|Spectral analysis of a stationary stochastic process]]). The regression spectrum for  $  M ( \lambda ) $
 
is the set of all  $  \lambda $
 
is the set of all  $  \lambda $

Latest revision as of 20:35, 16 January 2024


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=55155
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article