Consider a special case of the nested error regression model (14.85) with x



ijβ = μ and ni = k, 1 ≤ i ≤ m, where μ is an unknown mean and k ≥ 2. Suppose that the random effects vi are i.i.d. with an unknown distribution F that has mean 0 and finite moment of any order; the errors eij are also i.i.d. with mean 0 and finite moment of any order. The REML estimators of the variances σ2 v , σ2 e , which do not require normality (see Section 12.2), are given by ˆσ 2 v

= (MSA − MSE)/k and ˆσ 2 e

= MSE, where MSA = SSA/(m − 1) with SSA = k



m i=1(¯yi· − ¯y··)2, ¯yi· = k

−1



k j=1 yij ,

¯y

·

·

=

(

mk)

−1



m i=1



k j=1 yij , and MSE = SSE/m(k − 1) with SSE =



m i=1



k j=1(yij − ¯yi·)2, if one ignores the nonnegativity constraint on ˆσ 2 v , which you may throughout this exercise for simplicity. The EBLUP for the random effect vi is given by

ˆv i = kˆσ 2 v (ˆσ 2 e

+ kˆσ2 v )

−1(¯yi· − ¯y··).

Suppose that m→∞while k is fixed. Show that MSPE(ˆvi ) = E(ˆvi − vi )2 can be expressed as a + o(m

−1), where a depends only on the second and fourth moments of F and G.