CONSISTENCY OF ESTIMATORS IN FACTOR ANALYSIS
CONSISTENCY OF ESTIMATORS IN FACTOR ANALYSIS
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因子分析中估计量的一致性
DOI：
10.11329/jjss1970.13.137
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发表时间：
1983
期刊：
影响因子：
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通讯作者：
Y. Kano
中科院分区：
文献类型：
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作者：
Y. Kano
1.简介 在许多统计模型中，最大似然估计量（MLE）和广义最小二乘估计量（GLSE）在某些规律性条件下是一致的。例如，Rao [6]（第356-360页）展示了强可辨识条件下多项模型中MLE的一致性，在作者看来，这对于协方差结构模型的一致性也是必需的。在协方差结构模型的正态性假设下，强可辨识性意味着不存在远离真实结构参数的结构参数，从而产生几乎相同的真实方差矩阵。因此，Anderson 和 Rubin [1]（第 145-146 页）在因子分析模型中以及 Browne [2]（第 7 页）在协方差结构模型中的一致性证明可能不够严格，因为他们默认了强可识别性。另一方面，Tumura 和 Fukutomi [7]（第 68-69 页）和 Fukutomi [3]（第 130 页）指出，因子分析模型中的 MLE 相对于样本方差矩阵可能是不连续的，或者换句话说，估计量可能会因样本方差矩阵的微小变化而大幅波动，因此即使样本量为 大的。但 Ninomiya [4] 得出的结论是，MLE 的一致性不是通过分析推理，而是通过她特定模型中的模拟实验。 Okamoto 和 Ihara [5] 还通过实验提出了 Tumura 和 Fukutomi [7] 提出的数值模型中 LSE 的一致性。在本文中，假设Anderson和Rubin对真实方差矩阵的弱可辨识性的充分条件，我们首先证明在正态性假设下，作为样本方差矩阵的函数的MLE在真实方差矩阵处是连续的，并且类似地GLSE在那里是连续的，概率为1。那么，如果样本方差矩阵弱（或强）收敛于真实方差矩阵，则MLE弱（或强）一致，此外，如果权重矩阵弱（或强）收敛，则GLSE弱（或强）一致。
1. Introduction In many statistical models, both the maximum likelihood estimator (MLE) and the generalized least squares estimator (GLSE) are consistent under some regularity conditions. For example, Rao [6](pp. 356-360) shows the consistency of the MLE in a multinomial model under strong identifiability condition, which seems to the author to be required also for the consistency in covariance structure model. Under the normality assumption in covariance structure model, the strong identifiability means that there is no structural parameter remote from the true one which yields nearly the same true variance matrix. Thus, the proofs of the consistency by Anderson and Rubin [1](pp. 145-146) in a factor analysis model and by Browne [2](p. 7) in a covariance structure model might be short of com plete rigor in that they assume tacitly the strong identifiability. On the other hand, it is stated in Tumura and Fukutomi [7](pp. 68-69) and Fukutomi [3](p. 130) to the effect that the MLE in a factor analysis model may be discontinuous with respect to the sample variance matrix, or in other words the estimator can fluctuate wildly for small variations of the sample variance ma trix, so that the estimator cannot be reliable even if the sample size is large. But Ninomiya [4] drew a conclusion that the MLE is consitent not by analytical rea soning but by simulation experiments in her particular models. Okamoto and Ihara [5] suggested also experimentally the consistency of the LSE in the numerical models presented by Tumura and Fukutomi [7]. In this paper, assuming Anderson and Rubin's sufficient condition for weak identifiability for the true variance matrix, we show first that under the normality assumption the MLE which is a function of the sample variance matrix is con tinuous at the true variance matrix and that similarly the GLSE is continuous there with probability going to one. Then, if the sample variance matrix con verges to the true variance matrix weakly (or strongly), the MLE is consistent weakly (or strongly), and furthermore if the weight matrix converges weakly (or strongly), the GLSE is consistent weakly (or strongly).