F‐tests for Hypotheses with Block Matrices and Under Conditions of Orthogonality in the General Multivariate Gauss‐Markoff Model

The multivariate general Gauss‐Markoff (MGM) model ( U, XB , ∑⊗σ 2 V ) when the matrices V ≥ 0 and ∑ > 0 are known and the scalar σ 2 > 0 is unknown, is considered. The present paper is a continuation of two earlier works (Oktaba, 1988a, b). If XB = X 1 Σ + X 2 Δ, then the F‐test for verification the hypothesis W Σ A = 0 is presented. Moreover, under conditions of orthogonality the decomposition of the matrix S A (Ł BCA )′ L − (Ł BCA ) into the sum of s = r(L) matrices is given, where Ł BCA is the estimator of the parametric estimable functions Ł BCA , Cov (Ł BCA ) = A ′ ∑⊗σ 2 L = Ł C 4 Ł′, B̌ = ( X ′ T − X ) − X ′ T − U , C 4 = ( X ′ T − X ) − M , where M = M ′ is any arbitrary matrix such that R(X ) ⊂ R(T) , T = V + XMX ′; T − is any c ‐inverse. R(A) is the linear space generated by the colums of A . Then under additional assumption on normality of U the statistics F for testing Ł BA = 0 is deduced. Under conditions of normality of U and decomposition of S A , the statistics F 1 , …, F s for the hypotheses j i BA = 0 ( i = 1,…, s ) are established.