5.6 (Marshall and Mardia, 1985) This exercise develops the principle of MINQUE for certain spatial processes for which the mean vanishes and the covariance function is linear in the unknown parameters.

(a) let x ∼ Np(????, Ψ) where Ψ = ????2A + ????2I. Here, A is a known positivedefinite symmetric matrix from a parametric model, and the second term represents a nugget effect. The objective is to estimate the scaling parameters ????2 ≥ 0 and ????2 ≥ 0 from a single realization of the vector x.

See Mardia and Marshall (1984) for more details.

From the data, construct two statistics, u = xTx, and ???? = xTAx. Also set m0 = p, m1 = tr(A), m2 = tr(A2).

Recall for a general symmetric matrix B that E(xTBx) = E{tr(BxxT)} =

tr(BΨ). Hence, deduce that E(u) = ????2 m1 + ????2 m0, E(????) = ????2 m2 + ????2 m1.