Selection of Loss Function in Covariance Structure Analysis: Case of the Spherical Model

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.

KEYWORDS:

• Covariance structure analysis
• structural equation modeling
• factor analysis
• spherical model
• high-dimensional framework
• LR test

Acknowldgement

The authors are grateful to the two reviewers for helpful comments. The first author acknowledges financial support from the Grant-in-Aid for Scientific Research (KAKENHI 16H03606, 17K03660, 17KK0070, 20H01484) provided by the JSPS.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Supplementary material

Supplemental data for this article can be accessed on the publisher’s website

Notes

¹ We assume normality throughout the paper to simplify the derivation.

² Note that the original function is not divided by $p$. However, we use this definition because we consider the case in which both $n$ and $p$ are large.

³ For the derivation, see the appendix.

⁴ These functions are derived from Gupta and Nagar (1999, Theorem 3.3.15, and 3.3.16).

⁵ Since the ML, invGLS, and invGLSub do not use the inverse of the sample covariance matrix $\mathbf{S}{ }_n$, the condition $0<c<1$ is not necessary for these three estimators, and can be considered for the extended region $c\ge 1$. However, as there are no large differences if $c$ is extended to $c\ge 1$, we consider only $0<c<1$.

⁶ See, for example, Boomsma (1982), Hu et al. (1992), Bentler and Yuan (1999), Fouladi (2000), and Nevitt and Hancock (2004), and Herzog et al. (2007), and Moshagen (2012).

Additional information

Funding

This work was supported by the JSPS [KAKENHI 16H03606, 17K03660, 17KK0070, 20H01484, 20K20760].