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F-tests for hypotheses with block matrices and under conditions of orthogonality in the general multivariate Gauss-Markoff model. (English) Zbl 0712.62056

Summary: The multivariate general Gauss-Markoff (MGM) model \((U,XB,\Sigma \otimes \sigma^ 2V)\) when the matrices \(V\geq 0\) and \(\Sigma >0\) are known and the scalar \(\sigma^ 2>0\) is unknown, is considered. If \(XB=X_ 1\Gamma +X_ 2\Delta\), then the F-test for verifying the hypothesis \(W\Gamma A=0\) is presented. Moreover, under conditions of orthogonality the decomposition of the matrix \(S_ A=(\tilde L\hat BA)'L-(\tilde L\hat BA)\) into the sum of \(s=r(L)\) matrices is given, where \(\tilde L\hat BA\) is the estimator of the parametric estimable functions \(\tilde LBA\), \[ Cov(\tilde L\hat BA)=A'\Sigma A\otimes \sigma^ 2L,\quad L=\tilde LC_ 4\tilde L',\quad \hat B=(X'T^-X)^-X'T^-U,\quad C_ 4=(X'T^- X)^--M, \] where \(M=M'\) is any arbitrary matrix such that \(R(X)\subset R(T)\), \(T=V+XMX'\); \(T^-\) is any c-inverse. R(A) is the linear space generated by the columns of A. Then under additional assumption on normality of U the statistics F for testing \(\tilde LBA=0\) is deduced. Under conditions of normality of U and decomposition of \(S_ A\), the statistics \(F_ 1,...,F_ s\) for the hypotheses \({\tilde \ell}'_ iBA=0\) \((i=1,...,s)\) are established.

MSC:

62H15 Hypothesis testing in multivariate analysis
62J99 Linear inference, regression
62H12 Estimation in multivariate analysis
15A09 Theory of matrix inversion and generalized inverses
15A99 Basic linear algebra
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References:

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