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Best linear unbiased estimator

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BLUE

Let

(a1)

be a linear regression model, where is a random column vector of "measurements" , is a known non-random "plan" matrix, is an unknown vector of the parameters, and is a random "error" , or "noise" , vector with mean and a possibly unknown non-singular covariance matrix . A model with linear restrictions on can be obviously reduced to (a1). Without loss of generality, .

Let ; a linear unbiased estimator (LUE) of is a statistical estimator of the form for some non-random matrix such that for all , i.e., . A linear unbiased estimator of is called a best linear unbiased estimator (BLUE) of if for all linear unbiased estimators of , i.e., if for all linear unbiased estimators of and all .

Since it is assumed that , there exists a unique best linear unbiased estimator of for any . It is then given by the formula , where , which coincides by the Gauss–Markov theorem (cf. Least squares, method of) with the least square estimator of , defined as ; as usual, stands for transposition.

Because is normally not known, Yu.A. Rozanov [a2] has suggested to use a "pseudo-best" estimator in place of , with an appropriately chosen . 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 assumed to belong to an arbitrary known convex set of positive-definite -matrices with respect to the general quadratic risk function of the form

where is any non-negative-definite -matrix and stands for the expectation assuming . 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
How to Cite This Entry:
Best linear unbiased estimator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Best_linear_unbiased_estimator&oldid=12048
This article was adapted from an original article by I. Pinelis (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article