19.4 Nonidentical variances

The assumption of identical variances in Theorem 19.1 is sometimes not justified. Moreover the error terms described by stochastic quantities Ui are not always uncorrelated. Therefore the regression model

is generalized in the following way:

where ∑ is a given positive definite (n×n) matrix.

The corresponding linear regression model is

and including the error terms Ux it becomes

with VCov(U) = σ2 ∑.

Again the method of LSS is applied to estimate the parameters θ0, θ1,…,θr from data (xi, yi), i = 1(1)n. The following theorem is a generalization of the Gauss–Markov theorem, called the Gauss–Markov–Aitken theorem.

Theorem 19.5

Let a general linear model (19.9) be given. Furthermore a sample (xi Yi), i = 1(1)n. Assuming the following conditions are fulfilled:

1. rank

2. VCov (Y) = σ2 ∑;

3. the space of possible parameters θ is not a hyperplane in r + 1,

then ...

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