Robust factor analysis with exponential squared loss

The large dimensional factor model, aimed at reducing dimensionality and extracting features through a few latent common factors, has sparked significant interest due to its broad applications. Despite the popularity of traditional methods for factor models, they may yield incorrect estimators for heavy-tailed data. To address this issue, we introduce the exponential squared loss to the factor model in this study, denoted as the Robust Exponential Factor Analysis (REFA). We propose a modified rank minimization technique to enhance the estimation accuracy of factor numbers in finite-sample cases. Consistency properties for factors and loadings are established under mild conditions, without any moment assumptions on the errors. The performance of REFA with finite samples under both light and heavy-tailed cases has been demonstrated through simulation studies. Furthermore, an analysis employing a financial dataset of asset returns underscores the superiority of REFA. To facilitate the implementation of our proposed methodology by researchers, we have developed an R package named REFA, which is available on CRAN.