The additional assumption in part (ii) of Theorem 12.1 that the random effects and errors are nondegenerate is necessary for the asymptotic normality of the REML estimators. To see a simple example, suppose that yi = μ + i , i = 1, . . . , m, where the i ’s are independent such that P(i = −1) = P(i = 1) = 1/2. Note that in this case there are no random effects and the i ’s are the errors.

(i) Show that the REML estimator of the variance of the errors, σ2 = 1, is the sample variance, ˆσ 2 = (m − 1)

−1



m i=1(yi − ¯y)2, where ¯y =

m

−1



m i=1 yi .

(ii) Show that

√

m(ˆσ 2 − 1) does not converge in distribution to a (nondegenerate)

normal distribution.