Suppose that observations have been made at a sample of observed y-values at n sites t1, ..., tn. Write 2Yij = var(Y(ti) Y (t;)) var(Y(ti) Y(t;)). We want to predict the value of the random variable Y(to) at the site to through (to) where n (to) = ai (ti) i=1 The value of a1, ..., an can be determined by minimizing the mean square prediction error and also to yield unbiased (to) E (Y (to) - (to)) E (to) = EY (to) Show that the solution is the following where a1 a = - Y10 Y20 = Y11 Y12 1 V2n 1 721 722 -(0)-(0)-(0) a= m* = : 1 : : Yn2 1 1 1 2 Hint: Use the following relationship between Yi; and Cij = 2ij var(Y(ti) - Y(t;)) = var(Y(ti)) + var(Y(t;)) - 2cov(Y(t;), Y(t;)) = Cii + C'jj 2Cij 20 = var(Y(ti) Y(t;)) = var(Y(t;)) + var(Y(t;)) 2cov(Y(ti), Y(t;)) = Coo+Cii - 2C10 Incorporate these result into n n Coo+a;a; - n Coo + aia;Cij - 2 :Cio i=1 j=1 i=1 on page 5 of the "Kriging_theory" handout, and then carry out the remaining proof.