4.28 Consider a special case of the Fay–Herriot model (Example 4.18) in which Di = D, 1 ≤ i ≤ m. This is known as the balanced case. Without loss of generality, let D = 1. Consider the prediction of ηi = x



iβ + vi. Let ˜ηi and ˆηi denote the BLUP and EBLUP, respectively, where the MoM estimator of A is used for the EBLUP [see Example 4.18 (continued) or Exercise 4.25].

(i) Show that MSPE(˜ηi ) = A A + 1

+ x



i(X



X)

−1xi A + 1

, where X = (x



i )1≤i≤m.

(ii) Show that MSPE(ˆηi ) = A A + 1 + x 
i(X 
X)
−1xi A + 1 +
2{1 − x 
i(X 
X)
−1xi }
(A + 1)(m − p)
+
4{1 − x 
i(X 
X)
−1xi }
(A + 1)(m − p)(m − p − 2)
.
[Hint: The moment of (χ2 k )
−1 has a closed-form expression, where χ2 k denotes a random variable with a χ2 k -distribution. Find the expression.]
(iii) Let η = (ηi )1≤i≤m denote the vector of small-area means and ˆη = (ˆηi )1≤i≤m denote the vector of EBLUPs. Define the overall MSPE of the EBLUP as MSPE( ˆ η) = E(| ˆ η − η|2) = E{

m i=1(ˆηi − ηi )2  } = m i=1 E(ˆηi − ηi )2 = 
m i=1 MSPE(ˆηi ). Show that MSPE( ˆ η) = mA A + 1 + p + 2 A + 1 + 4 (A + 1)(m − p − 2)
.