The restricted maximum likelihood (REML) estimates of dispersion param
eters (variance components) in a general (non-normal) mixed model are
defined as solutions of the REML equations. In this paper, we show the
REML estimates are consistent if the model is asymptotically identifi
able and infinitely informative under the (location) invariant class,
and are asymptotically normal (A.N.) if in addition the model is asymp
totically nondegenerate. The result does not require normality or boun
dedness of the rank p of design matrix of fixed effects. Moreover, we
give a necessary and sufficient condition for asymptotic normality of
Gaussian maximum likelihood estimates (MLE) in non-normal cases. As an
application, we show for all unconfounded balanced mixed models of th
e analysis of variance the REML (ANOVA) estimates are consistent; and
are also A.N. provided the models are nondegenerate; the MLE are consi
stent (A.N.) if and only if certain constraints on p are satisfied.