Selecting a Shrinkage Parameter in Structural Equation Modeling with a near Singular Covariance Matrix by the Gic Minimization Method

In structural equation modeling (SEM), a covariance parameter is derived by minimizing the discrepancy between a sample covariance matrix and a covariance matrix having a specified structure. When a sample covariance matrix is a near singular matrix, Yuan and Chan (2008) proposed the use of an adjusted sample covariance matrix instead of the sample covariance matrix in the discrepancy function for estimating the covariance estimator. The adjusted sample covariance matrix was defined by adding an identity matrix multiplied by a shrinkage parameter to the existing sample covariance matrix. They used a constant value as the shrinkage parameter, which was chosen based solely on the sample size and the number of dimensions of the observation, and not on the data itself. However, selecting the shrinkage parameter from the data may lead to a greater improvement in prediction compared to the use of a constant shrinkage parameter. Hence, we attempt to select the shrinkage parameter using an information criterion minimization method. Therefore, we propose an information criterion based on the discrepancy function measured by the normal theory maximum likelihood (ML). Using the Monte Carlo method, we demonstrate that the proposed criterion works well.