Prove three facts about Kriging.

(a) Show that if b'Y is the BLUP of yo and if Xi! = 1, i = 0,1, ... , n, then b'J = 1.

(b) Show that if (yO, x~) = (Yi, x~) for some i ~ 1, then the BLUP of yo is just Yi'

(c) Show that if V is nonsingular and b'Y is the BLUP of yo, then there exists a vector 'Y such that the following equation is satisfied:

Typically, this equation will have a unique solution.

Hint: Recall that x~ = p' X and that X(X'V- 1 X)- X'V- 1 is a projection operator onto C(X).

EXAMPLE 12.2.6. Mixed Model Prediction.

In the mixed model (12.1.1), we wish to find the BLUP of )..''Y based on the data Y. The vector).. can be any known vector. Note that E()..''Y) = OJ so we can let yo = >.''Y and x~ = O. From Section 1, V = ZDZ' + Rand VyO = Cov(Y, )..' 'Y)

Cov(Z'Y + e,>.''Y)

= Cov(Z'Y, )..''Y)

= ZD)".