In this paper, we consider the problem of selecting the variables of the fixed
effects in the linear mixed models where the random effects are present and the
observation vectors have been obtained from many clusters. As the variable selection
procedure, we here use the Akaike Information Criterion, AIC. In the context of
the mixed linear models, two kinds of AIC have been proposed: marginal AIC
and conditional AIC. In this paper, we derive three versions of conditional AIC
depending upon different estimators of the regression coefficients and the random
effects. Through the simulation studies, it is shown that the proposed conditional
AIC's are superior to the marginal and conditional AIC's proposed in the literature
in the sense of selecting the true model. Finally, the results are extended to the
case when the random effects in all the clusters are of the same dimension but have
a common unknown covariance matrix.
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