The Akaike information criterion, AIC, and Mallows' Cp statistic have been
proposed for selecting a smaller number of regressor variables in the multivariate
regression models with fully unknown covariance matrix. All these criteria are,
however, based on the implicit assumption that the sample size is substantially
larger than the dimension of the covariance matrix. To obtain a stable estimator
of the covariance matrix, it is required that the dimension of the covariance matrix
be much smaller than the sample size. When the dimension is close to the sample
size, it is necessary to use ridge type of estimators for the covariance matrix. In
this paper, we use a ridge type of estimators for the covariance matrix and obtain
the modified AIC and modified Cp statistic under the asymptotic theory that both
the sample size and the dimension go to infinity. It is numerically shown that these
modified procedures perform very well in the sense of selecting the true model in
large dimensional cases.
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