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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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