  Given a random sample of size $n$ from a $p$ dimensional random vector, where
both $n$ and $p$ are large, we are interested in testing whether the $p$
components of the random vector are mutually independent. This is the so-called
complete independence test. In the multivariate normal case, it is equivalent
to testing whether the correlation matrix is an identity matrix. In this paper,
we propose a one-sided empirical likelihood method for the complete
independence test for multivariate normal data based on squared sample
correlation coefficients. The limiting distribution for our one-sided empirical
likelihood test statistic is proved to be $Z^2I(Z>0)$ when both $n$ and $p$
tend to infinity, where $Z$ is a standard normal random variable. In order to
improve the power of the empirical likelihood test statistic, we also introduce
a rescaled empirical likelihood test statistic. We carry out an extensive
simulation study to compare the performance of the rescaled empirical
likelihood method and two other statistics which are related to the sum of
squared sample correlation coefficients.
