Robust estimation in structural equation models using Bregman and other divergences with t-centre approach to estimate the covariance matrix

Abstract

Structural equation models seek to find causal relationships between latent variables by analysing the mean and the covariance matrix of some observable indicators of the latent variables. Under a multivariate normality assumption on the distribution of the latent variables and of the errors, maximum likelihood estimators are asymptotically efficient. The estimators are significantly influenced by violation of the normality assumption and hence there is a need to robustify the inference procedures. Previous work minimized the von Neuman divergence or its variant the total von Neumann divergence to estimate the parameters, with the minimum covariance determinant used as a robust estimator of the covariance matrix. We extend this approach by considering other divergences and by developing a robust estimate of the covariance matrix. The robust estimator of the covariance matrix developed is a t-centre like estimator based on several minimum covariance determinant estimators ranging from 0% contamination to 50% contamination. The simulation results are promising. The results can be used for robustifying the fit of structural equation models. References K. A. Bollen. Structural equations with latent variables . Wiley, New York, 1989. http://au.wiley.com/WileyCDA/WileyTitle/productCd-0471011711.html R. W. Butler, P. L. Davies and M. Jhun. Asymptotics for the minimum covariance determinant estimator. Ann. Stat. , 21(3):1385–1400, 1993. http://www.jstor.org/stable/2242201 N. J. Higham. Functions of matrices: Theory and computation. SIAM, Philadelphia, 2008. doi:10.1137/1.9780898717778 S. Jayasumana, R. Hartley, M. Salzmann, H. Li and M. Harandi. Kernel methods on the Riemannian manifold of symmetric positive definite matrices. Proc. CVPR IEEE 2013 , 73–80, 2013. doi:10.1109/CVPR.2013.17 S. Penev and T. Prvan. Robust estimation in structural equation models using Bregman divergences. CTAC2012, ANZIAM J. , 54:C574–C589, 2013. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6306 B. C. Vemuri, M. Liu, S.-I. Amari, and F. Nielsen. Total Bregman divergence and its applications to DTI analysis. IEEE T. Med. Imaging. , 30(2):475–483, 2011. doi:10.1109/TMI.2010.2086464 S. Verboven and M. Hubert. LIBRA: a MATLAB library for robust analysis, Chemometr. Intell. Lab. , 75(2):127–136, 2005. doi:10.1016/j.chemolab.2004.06.003