AI 中文总结
该研究提出用Bures-Wasserstein公式化矩阵分解结构方程模型，建立其与协方差结构方程模型的等价关系，证明了估计量的理论性质，且其有限样本性能、数值稳定性优于传统估计量。
AI 中文摘要
矩阵分解结构方程模型（MDSEM）是一种基于数据矩阵的、替代传统协方差结构方程模型（SEM）的方法，但其与协方差SEM的理论关系以及其估计量的统计性质尚不明确。我们首先使用整合了测量模型和结构模型的单一损失函数重新公式化MDSEM。随后，我们证明在模型参数上最小化该损失函数等价于最小化观测协方差矩阵与模型隐含协方差矩阵之间的平方Bures-Wasserstein（BW）距离，并利用该等价关系建立估计量的理论性质。具体而言，该等价关系表明所提出的估计量既是MDSEM估计量，也是最小差异估计框架内的协方差SEM估计量，由此可推导其一致性、渐近正态性和标准误。模拟实验显示，其有限样本性能与包括最大似然法在内的传统估计量相当，且置信区间覆盖率接近标称水平；在小样本及模型误设情况下，该估计量比传统SEM估计量具有更高的数值稳定性。因此，BW公式化为MDSEM提供了理论基础，并为协方差SEM提供了实用的差异函数。
英文摘要
Matrix decomposition SEM (MDSEM) is a data-matrix-based alternative to conventional covariance-based SEM, but its theoretical relationship to covariance-based SEM and the statistical properties of its estimator have remained unclear. We first reformulate MDSEM using a single loss function that integrates the measurement and structural models. We then prove that minimizing this loss over the model parameters is equivalent to minimizing the squared Bures-Wasserstein (BW) distance between the observed and model-implied covariance matrices, and use this equivalence to establish the estimator's theoretical properties. Specifically, the equivalence identifies the proposed estimator as both an MDSEM estimator and a covariance-based SEM estimator within the minimum discrepancy estimation framework, from which consistency, asymptotic normality, and standard errors are derived. Simulations show that its finite-sample performance is comparable to that of conventional estimators, including maximum likelihood, and that confidence-interval coverage is close to the nominal level. The estimator is also more numerically stable than conventional SEM estimators in small samples and under model misspecification. Thus, the BW formulation provides a theoretical foundation for MDSEM and a practically useful discrepancy function for covariance-based SEM.
