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Additional resources for An introduction to multicomplex spaces and functions
G0 G/ 1 X0 y. G0 G/ 1 X0 y D P˝ y. G0 G/ 1 X0 . 3)). 1) holds for β, 0 H are linearly independent. G0 G/ 1 X0 D 0, and this completes the theorem. G0 G/ 1 G0 Gβ D β EŒβ and βO is unbiased. 6) in Sect. 10). without changing ˝ (as indicated by λ However, another method of finding a solution is to use a weak (generalized) inverse of X0 X. A weak inverse of a matrix L is any matrix L satisfying LL L D L (See Seber 2008, chapter 7). X0 X/ X0 y is a solution of the normal equations. 2). G0 G/ 1 is a weak inverse of X0 X.
All we require is θ, for testing H is FD Minimizing i . 5 Confidence Regions and Intervals In most practical applications of linear hypothesis theory our prime interest is not just in significance tests but also in the finding of confidence regions and confidence intervals for the unknown parameters.
1, and we wish to test Aβ D b, where A is q p of rank q. Let β0 be any solution of Aβ D b, put z D y Xβ0 and let γ D β β0 . Then our original model and hypothesis are equivalent to z D Xγ C ε, where ε is Nn Œ0; 2 In , and ! W Aγ D 0. X0 X/ 1 X0 φ, ! X0 X/ 1 X0 . X0 X/ 1 A0 c D 0, which pre-multiplying by X0 give us A0 c D 0 or c D 0, as the q columns of A0 are linearly independent. Hence the rows of A1 are linearly independent and A1 has rank q. Also, CŒA01 \ ˝ ? CŒX \ ˝ ? 7) applies with A1 P˝ D A1 (since P˝ X D X).