# zbMATH — the first resource for mathematics

Linear estimation with an incorrect dispersion matrix in linear models with a common linear part. (English) Zbl 0536.62052
The linear model (Y,$$X\beta$$,V) where $$Y=X\beta +e$$ and $$V=(cov(e))$$ is considered. For any matrix A, Let $${\mathcal R}(A)$$ denote the vector space spanned by the columns (or the range space); r(A) denotes the rank of A. The article starts with the set up considering all linear models $$(Y,X\beta,V_ 1)$$ with a common linear part $${\mathcal R}(U)$$, that is $$r(X)=p<n$$ such that $${\mathcal R}(U)\subset {\mathcal R}(X)$$, U being a known matrix with $$r(U)<p$$. The class of such matrices is denoted by $$C^ p(U)$$. The considered design matrices $$X\in C^ p(U)$$ further satisfy the condition $${\mathcal R}(A^ t)\subset {\mathcal R}(X^ t)$$, where A is a given matrix with 1$$\leq r(A)\leq p$$. The class of such matrices is denoted by $$C_ A^ p(U).$$
The author characterizes matrices V such that every linear representation or some linear representation of the BLUE of $$A\beta$$ under $$(Y,X\beta,V_ 1)$$ continues to be its BLUE under (Y,$$X\beta$$,V) also. Further on, variance components and covariance models are dealt with. Three conditions are proved under which $$A\beta$$ admits a BLUE for every $$X\in C_ A^ p(U)$$.
Reviewer: B.Kolleck

##### MSC:
 62J05 Linear regression; mixed models 62J10 Analysis of variance and covariance (ANOVA)
Full Text: