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Linear model with variances depending on the mean value. (English) Zbl 0764.62055
Let \((Y,X\beta,\Sigma)\) be a linear regression model. The result of the observations is a realization of a random vector \(Y_{n,1}\), whose mean value is \(E_ \beta Y=X\beta\), \(X_{n,k}\) is a known design matrix, \(\beta_{k,1}\in R^ k\) the vector of unknown parameters, the covariance matrix of the vector \(Y\) depends on \(\beta\), \[ \Sigma=\sigma^ 2\Sigma(\beta)=\text{diag}(\sigma^ 2(a+b| e_ i' X\beta|^ 2))_{1\leq i\leq n}, \] where \(\sigma^ 2\), \(a\) and \(b\) are known positive constants, and \(e_ i'\) is the transpose of the \(i\)th unity vector.
The \(\beta_ 0\)-locally best linear unbiased estimator of a linear function of the parameter \(\beta\) is obtained.
Reviewer: N.Leonenko (Kiev)

MSC:
62J05 Linear regression; mixed models
62H12 Estimation in multivariate analysis
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References:
[1] KUBÁČEK L.: Foundations of Estimation Theory. Elsevier, Amsterdam-Oxford-NewYork-Tokyo, 1988. · Zbl 0698.62004
[2] RAO C. R., MITRA S. K: Generalized Inverse of Matrices and its Applications. J.Wiley, New York, 1971. · Zbl 0236.15005
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