View source for Best linear unbiased estimator

<!--
b1104401.png
$#A+1 = 48 n = 0
$#C+1 = 48 : ~/encyclopedia/old_files/data/B110/B.1100440 Best linear unbiased estimator,
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}}

''BLUE''

Let

$$ \tag{a1 }
Y = X \beta + \epsilon
$$

be a [[Linear regression|linear regression]] model, where  $  Y $
is a random column vector of  $  n $ "measurements" ,  $  X \in \mathbf R ^ {n \times p } $
is a known non-random  "plan"  matrix,  $  \beta \in \mathbf R ^ {p \times1 } $
is an unknown vector of the parameters, and  $  \epsilon $
is a random  "error" , or  "noise" , vector with mean  $  {\mathsf E} \epsilon =0 $
and a possibly unknown non-singular covariance matrix  $  V = { \mathop{\rm Var} } ( \epsilon ) $. 
A model with linear restrictions on  $  \beta $
can be obviously reduced to (a1). Without loss of generality,  $  { \mathop{\rm rank} } ( X ) = p $.

Let  $  K \in \mathbf R ^ {k \times p } $; 
a linear unbiased estimator (LUE) of  $  K \beta $
is a [[Statistical estimator|statistical estimator]] of the form  $  MY $
for some non-random matrix  $  M \in \mathbf R ^ {k \times n } $
such that  $  {\mathsf E} MY = K \beta $
for all  $  \beta \in \mathbf R ^ {p \times1 } $, 
i.e.,  $  MX = K $. 
A linear [[Unbiased estimator|unbiased estimator]]  $  M _ {*} Y $
of  $  K \beta $
is called a best linear unbiased estimator (BLUE) of  $  K \beta $
if  $  { \mathop{\rm Var} } ( M _ {*} Y ) \leq  { \mathop{\rm Var} } ( MY ) $
for all linear unbiased estimators  $  MY $
of  $  K \beta $, 
i.e., if  $  { \mathop{\rm Var} } ( aM _ {*} Y ) \leq  { \mathop{\rm Var} } ( aMY ) $
for all linear unbiased estimators  $  MY $
of  $  K \beta $
and all  $  a \in \mathbf R ^ {1 \times k } $.

Since it is assumed that  $  { \mathop{\rm rank} } ( X ) = p $, 
there exists a unique best linear unbiased estimator of  $  K \beta $
for any  $  K $. 
It is then given by the formula  $  K {\widehat \beta   } $, 
where  $  {\widehat \beta   } = { {{\widehat \beta   } _ {V} } } = ( X  ^ {T} V ^ {-1 } X ) ^ {-1 } X  ^ {T} V ^ {-1 } Y $, 
which coincides by the Gauss–Markov theorem (cf. [[Least squares, method of|Least squares, method of]]) with the least square estimator of  $  \beta $, 
defined as  $  { \mathop{\rm arg} } { \mathop{\rm min} } _  \beta  ( Y - X \beta )  ^ {T} V ^ {- 1 } ( Y - X \beta ) $; 
as usual,  $  {}  ^ {T} $
stands for transposition.

Because  $  V = { \mathop{\rm Var} } ( \epsilon ) $
is normally not known, Yu.A. Rozanov [[#References|[a2]]] has suggested to use a  "pseudo-best"  estimator  $  { {{\widehat \beta   } _ {W} } } $
in place of  $  { {{\widehat \beta   } _ {V} } } $, 
with an appropriately chosen  $  W $. 
This idea has been further developed by A.M. Samarov [[#References|[a3]]] and I.F. Pinelis [[#References|[a4]]]. In particular, Pinelis has obtained duality theorems for the minimax risk and equations for the minimax solutions  $  V $
assumed to belong to an arbitrary known convex set  $  {\mathcal V} $
of positive-definite  $  ( n \times n ) $-
matrices with respect to the general quadratic risk function of the form

$$ 
R ( V,W ) = {\mathsf E} _ {V} ( {\widehat \beta   } _ {W} - \beta )  ^ {T} S ( {\widehat \beta   } _ {W} - \beta ) ,
$$

$$ 
V \in {\mathcal V},  W \in {\mathcal V},
$$

where  $  S $
is any non-negative-definite  $  ( p \times p ) $-matrix and  $  {\mathsf E} _ {V} $
stands for the expectation assuming  $  { \mathop{\rm Var} } ( \epsilon ) = V $. 
Asymptotic versions of these results have also been given by Pinelis for the case when the  "noise"  is a second-order stationary stochastic process with an unknown spectral density belonging to an arbitrary, but known, convex class of spectral densities and by Samarov in the case of contamination classes.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  C.R. Rao,   "Linear statistical inference and its applications" , Wiley  (1965)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  Yu.A. Rozanov,   "On a new class of estimates" , ''Multivariate Analysis'' , '''2''' , Acad. Press  (1969)  pp. 437–441</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  A.M. Samarov,   "Robust spectral regression"  ''Ann. Math. Stat.'' , '''15'''  (1987)  pp. 99–111</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  I.F. Pinelis,   "On the minimax estimation of regression"  ''Th. Probab. Appl.'' , '''35'''  (1990)  pp. 500–512</TD></TR>
</table>