  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.
