Sparse Generalized Factor Models With Weaker Loadings

ABSTRACT Generalized factor models are gaining traction for multivariate data dimension reduction due to their flexibility. This paper investigates the sparse estimation of generalized factor models with weaker loadings. We introduce a group‐wise penalized estimation approach, which results in a sparse loading matrix. This sparsity not only facilitates variable selection but also enhances the interpretability of the reduced‐dimensional results. To tackle computational challenges, we develop a projected alternating maximization algorithm, achieving simultaneous parameter estimation and variable selection. Considering the importance of determining the number of factors, we propose a sparsity information criterion for this purpose. Under the weak loadings assumption and other mild conditions, we establish upper and lower error bounds for the overall parameter estimates, derive the convergence rates for the loading matrix and factor scores, and demonstrate the consistency of both variable selection and the number of factors. Furthermore, we extend the model to accommodate missing data, providing corresponding theoretical guarantees. The efficacy of our proposed method is validated through extensive simulations and applications to two real‐world datasets.