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Estimation of covariance components in a repeated regression experiment. (English) Zbl 0597.62052
In the regression model \(Y=X\beta +\epsilon\) the covariance matrix of the error vector \(\epsilon\) is considered in the form \(\Sigma =\sum^{m}_{i=1}J_ iCJ_ i'\); (n\(\times s)\)-matrices \(J_ i\), \(i=1,...,m\) are known. The elements of the unknown matrix C are called covariance components. When \(s=1\) and \(J_ iJ_ i'\) is denoted \(V_ i\), \(i=1,...,m\), the situation studied in the above reviewed paper, Zbl 0597.62051, occurs. This paper completes the above reviewed one. The aim is to determine the estimator of the covariance components on the basis of the matrix S, \[ kS=\sum^{k+1}_{j=1}(Y_ j-\bar Y)(Y_ j-\bar Y)'\quad (\bar Y=[1/(k+1)]\sum^{k+1}_{j=1}Y_ j), \] which is generated from the \((k+1)\)-tuple stochastically independent random vectors \(Y_ 1,...,Y_{k+1}\) with the same normal distribution \(N_ n(X\beta,\Sigma)\). Thus the matrix kS has the Wishart distribution \(W_ n(k,\Sigma)\).

62H12 Estimation in multivariate analysis
62J10 Analysis of variance and covariance (ANOVA)
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