5.16 Consider a nested three-way layout with Xijkl = + i + bij + cijk +...

5.16 Consider a nested three-way layout with Xijkl = µ + αi + bij + cijk + Uijkl

(i = 1,...,I ; j = 1,...,J ; k = 1,...,K; l = 1,...,n) in the versions

(a) ai = αi, bij = βij , cijk = γijk ;

(b) ai = αi, bij = βij , cijk = Cijk ;

(c) ai = αi, bij = Bij , cijk = Cijk ;

(d) ai = Ai, bij = Bij , cijk = Cijk ;

where the α’s, β’s, and γ ’s are unknown constants defined uniquely by the usual conventions, and the A’s, B’s, C’s, and U’s are unobservable random variables, independently normally distributed with means zero and with variances σ2 A, σ2 B, σ2 C and σ2.

In each case, transform the Xijkl to independent variables Zijkl and obtain the UMVU estimators of the unknown parameters.