Best linear unbiased estimator

BLUE

Let

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

be a 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 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 $ 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) 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 [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 [a3] and I.F. Pinelis [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

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