View source for Regression spectrum

<!--
r0806501.png
$#A+1 = 19 n = 0
$#C+1 = 19 : ~/encyclopedia/old_files/data/R080/R.0800650 Regression spectrum
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}}

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

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

where  $  x _ {t} $
is a [[Stationary stochastic process|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|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|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====
<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>