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On misspecification of the covariance matrix in linear models. (English) Zbl 1154.62050
Summary: For the true model denoted by $$\mathbf y = \mathbf X\pmb\beta+\mathbf e$$ with $$\mathbf e\sim {\mathcal N}(\mathbf 0,\sigma^2\Sigma)$$ and the misspecified model given as $$\mathbf y =\mathbf X\pmb\beta+\mathbf e$$ with $$\mathbf e \sim {\mathcal N}(\mathbf 0,\sigma^2 I_n)$$, the problem of estimating a parametric function, say $$\theta =\pmb\beta'H\pmb\beta + h\sigma^2$$, is considered. A general condition for the two best quadratic estimators for $$\theta$$ under true and misspecified models to be equal is investigated. Moreover, we apply our main results to a linear regression model with the covariance matrix being compound symmetric and derive an anticipant conclusion.

##### MSC:
 62J05 Linear regression; mixed models 62C15 Admissibility in statistical decision theory 62H12 Estimation in multivariate analysis
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