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On variance components estimation in heteroscedastic models. (English) Zbl 0614.62084

Statistical data which have been collected over a long period of time may show considerable lack of homogeneity in their precision. This causes usual statistical analysis to fail and often much less accurate measurements receive the same attention as newly obtained observations based on improved technology. Such experience invited the investigation of slightly generalized mixed linear models \(Y=X\beta +U_ 1\xi_ 1+...+U_ p\xi_ p\) where the covariance matrices of random effects vectors \(\xi_ i\) are no longer a multiple of the identity matrix, but at least for some i, they are diagonal matrices of heteroscedastic variances \(\sigma_{ij}\), \(j=1,...,n_ i.\)
We indicate the necessary changes of a numerical algorithm for deriving best linear estimators and predictors of fixed and random effects, respectively, and computation of minimum norm quadratic unbiased estimators (MINQUE) for the variances \(\sigma_{ij}\). Our method establishes the normal equations (analysis of variance table) without storing and inverting of matrices for all unbalanced nested classification models and extends to multivariate models at very little extra cost of computation. Some details of a corresponding numerical algorithm are presented and reference is made to an interactive computer program AMP 84.
Two examples are given to illustrate the way of computation and hint how to develop explicit expressions for the above mentioned estimates in simple cases.

MSC:

62J10 Analysis of variance and covariance (ANOVA)
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