Consider the correlated regression model, defined in the text by (5.58), say, y = Z + x, where x has mean zero and covariance matrix

Consider the correlated regression model, defined in the text by (5.58), say, y = Zβ + x, where x has mean zero and covariance matrix Γ. In this case, we know that the weighted least squares estimator is (5.59), namely,

βbw = (Z0

Γ −1Z)

−1Z0

Γ −1y.

Now, a problem of interest in spatial series can be formulated in terms of this basic model. Suppose yi = y(σi), i = 1, 2, . . . , n is a function of the spatial vector coordinates σi = (si1, si2, . . . , sir)0

, the error is xi = x(σi), and the rows of Z are defined as z(σi)0

, i = 1, 2, . . . , n. The Kriging estimator is defined as the best spatial predictor of y0 = z0 0β + x0 using the estimator yb0 = a0 y, subject to the unbiased condition Eyb0 = Ey0, and such that the mean square prediction error MSE = E[(y0 − yb0)

2]

is minimized.

(a) Prove the estimator is unbiased when Z0 a = z0.

(b) Show the MSE is minimized by solving the equations

Γ a + Zλ = γ0 and Z0 a = z0, where γ0 = E[xx0] represents the vector of covariances between the error vector of the observed data and the error of the new point the vector λ is a q × 1 vector of LaGrangian multipliers.

(c) Show the predicted value can be expressed as yb0 = z0 0βbw + γ0 0Γ −1(y − Zβbw), so the optimal prediction is a linear combination of the usual predictor and the least squares residuals.

Section 5.7