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).


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