Statistical properties of matrix decomposition factor analysis

For factor analysis, numerous estimators have been developed, and most of their statistical properties are well understood. In the early 2000s, a novel estimator based on matrix factorization, called Matrix Decomposition Factor Analysis (MDFA for short), was developed. Unlike classical estimators in factor analysis, the MDFA estimator offers several advantages, including the guarantee of proper solutions (i.e., no Heywood cases). However, the MDFA estimator cannot be formulated as a classical M-estimator or a minimum discrepancy function estimator, and the statistical properties of the MDFA estimator have yet to be discussed. Although the MDFA estimator is obtained by minimizing a principal component analysis-like loss function, it empirically behaves more like other consistent estimators for factor analysis than principal component analysis. We have an important question: Can matrix decomposition factor analysis truly be called "factor analysis"? To address this problem, we establish consistency and asymptotic normality of the MDFA estimator. Notably, the MDFA estimator can be formulated as a semiparametric profile likelihood estimator, and we derive the explicit form of the profile likelihood. Combined with its statistical properties, the empirical performance in the numerical experiments suggests that the MDFA estimator is a promising choice for factor analysis.