Selecting a shrinkage parameter in structural equation modeling with a near singular covariance matrix by the GIC minimization method

Abstract

In the structural equation modeling, unknown parameters of a covariance matrix are 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 estimation method to use an adjusted sample covariance matrix instead of the sample covariance matrix in the discrepancy function. The adjusted sample covariance matrix is defined by adding a scalar matrix with 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 propose an information criterion for selecting the shrinkage parameter, and attempt to select the shrinkage parameter by an information criterion minimization method. The proposed information criterion is based on the discrepancy function measured by the normal theory maximum likelihood. Using the Monte Carlo method, we demonstrate that the proposed criterion works well in the sense that the prediction accuracy of an estimated covariance matrix is improved.