Consistency of matrix decomposition factor analysis

For factor analysis, many estimators, starting with the maximum likelihood estimator, have been developed, and the statistical properties of most estimators have been well explored. In the early 2000s, a new estimator based on matrix factorization, called Matrix Decomposition Factor Analysis (MDFA), was developed. Although the estimator is obtained by minimizing the principal component analysis-like loss function, this estimator empirically behaves like other consistent estimators of factor analysis, not principal component analysis. Since the MDFA estimator cannot be formulated as a classical M-estimator, the statistical properties of the MDFA estimator have yet to be discussed. To explain this unexpected behavior theoretically, we establish the consistency of the MDFA estimator for factor analysis. That is, we show that the MDFA estimator converges to the same limit as other consistent estimators of factor analysis.