In the problem of selecting the explanatory variables in the linear mixed model,
we address the derivation of the (unconditional or marginal) Akaike information
criterion (AIC) and the conditional AIC (cAIC). The covariance matrices of the
random effects and the error terms include unknown parameters like variance components,
and the selection procedures proposed in the literature are limited to the
cases that the parameters are known or partly unknown. In this paper, AIC and
cAIC are extended to the situation that the parameters are completely unknown
and they are estimated by the general consistent estimators including the maximum
likelihood (ML), the restricted maximum likelihood (REML) and other unbiased estimators.
Related to AIC and cAIC, we derive the marginal and the conditional
prediction error criteria which select superior models in light of minimizing the prediction
errors relative to quadratic loss functions. Finally, numerical performances
of the proposed selection procedures are investigated through simulation studies.
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