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Bainbridge (1963), Staggered nested designs for estimating variance components, in American Society for Quality Control Annual Conference Transactions, American Society for Quality Control, Milwaukee, 93–103. R. B. Bapat and T. E. S. Raghavan (1997), Nonnegative Matrices and Applications, Cambridge University Press, Cambridge, UK. A. Basilevsky (1983), Applied Matrix Algebra in the Statistical Sciences, North-Holland, Amsterdam. A. Ben-Israel and T. Greyville (1974), Generalized Inverses: Theory and Applications, Wiley, New York.

4). 2) where β = (β1 , β2 ), without any consideration as to whether they represent fixed or random effects. 2). 2). 1, we have E(Y QY ) = E(Y )QE(Y ) + tr[Q Var(Y )]. 2), we obtain . E(Y ) = X1 E(β1 ) + X2 E(β2 ) = (X1 .. 3. Henderson’s Method III . = (X1 .. X2 ) X1 X2 Var(β1 ) Var(β2 ) + σe2 IN = X Var(β)X + σe2 IN . 4), we obtain E(Y QY ) = E(β )X QXE(β) + tr[Q{X Var(β)X + σe2 IN }] = tr[X QXE(ββ )] + σe2 tr(Q). 7) is true, irrespective of whether β is fixed or random. 2). 1), we have R(β1 , β2 ) = Y X(X X)− X Y .

Johnson (1985), Matrix Analysis, Cambridge University Press, Cambridge, UK. H. O. Lancaster (1954), Traces and cumulants of quadratic forms in normal variables, J. Roy. Statist. Soc. Ser. B, 16, 247–254. Bibliography 11 W. Madow (1940), The distribution of quadratic forms in noncentral normal random variables, Ann. Math. , 11, 100–101. J. R. Magnus and H. , Wiley, Chichester, UK. H. D. Muse and R. L. Anderson (1978), Comparison of designs to estimate variance components in a two-way classification model, Technometrics, 20, 159–166.

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