ABSTRACT

In this paper, we derive the asymptotic properties of estimators obtained from various kinds of loss functions in covariance structure analysis. We first show that the estimators except for OLS-based loss functions have the same asymptotic distribution when the dimension of the covariance matrix, $p$, is fixed and the sample size $n$ tends to infinity. Then, focusing on the spherical model, we show that this equivalence does not hold when both $n$ and $p$ become larger. Specifically, we show that some estimators lose consistency, and even consistent estimators have different asymptotic variances. Among the estimators considered, the maximum likelihood estimator shows the best performance, while the less famous invGLS(ub) estimator performs better than the commonly used GLS estimator. We also demonstrate the validity of the likelihood ratio test for the spherical and diagonal models in a high-dimensional framework.

摘要

在本文中，我们推导了从协方差结构分析中的各种损失函数获得的估计量的渐近性质。我们首先表明，当协方差矩阵的维数为$p$, 是固定的，样本量$n$趋于无穷大。然后，关注球面模型，我们证明当两者都存在时，这种等价性不成立$n$和$p$变大。具体来说，我们展示了一些估计量失去一致性，甚至一致的估计量也有不同的渐近方差。在考虑的估计器中，最大似然估计器表现出最好的性能，而不太知名的 invGLS(ub) 估计器比常用的 GLS 估计器表现更好。我们还证明了高维框架中球形和对角线模型的似然比检验的有效性。