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Asymptotical confidence region in a replicated mixed linear model with an estimated covariance matrix. (English) Zbl 0662.62056
Let $$Y_ 1,Y_ 2,..$$. be independent identically distributed (i.i.d.) random vectors; $$Y_ j\sim N_ n(X\beta,\Sigma)$$, $$j=1,2,...$$; the $$n\times k$$ matrix X is known, $$\beta\in {\mathcal R}^ k$$ (k-dimensional Euclidean space) is an unknown vector parameter and the covariance matrix $$\Sigma$$ is totally or partially unknown. The aim of the paper is to find a confidence region when some a priori information on the covariance matrix is available; we shall investigate two following cases:
a) the covariance matrix $$\Sigma$$ is diagonal with unknown elements;
b) the covariance matrix has the following structure: $\Sigma =\sum^{p}_{i=1}\vartheta_ iV_ i,\quad p\geq 2,\quad \vartheta =(\vartheta_ 1,...,\vartheta_ p)'\in {\underline \Theta}\subset {\mathcal R}^ p\quad ({\underline \Theta}\quad is\quad an\quad open\quad and\quad bounded\quad set),$ where the $$n\times n$$ symmetric matrices $$V_ i$$, $$i=1,...,p$$, are known and the components $$\vartheta_ 1,...,\vartheta_ p$$ are unknown (a mixed linear model).

MSC:
 62H12 Estimation in multivariate analysis 62F25 Parametric tolerance and confidence regions
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References:
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