### Robust estimation in structural equation models using Bregman divergences

#### 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. We propose to minimise the Bregman divergence or its variant, the total Bregman divergence, between a robust estimator of the covariance matrix and the model covariance matrix, with respect to the parameters of interest. Our approach to robustification is different from the standard approaches in that we propose to achieve the robustification on two levels: firstly, choosing a robust estimator of the covariance matrix; and secondly, using a robust divergence measure between the model covariance matrix and its robust estimator. We focus on the (total) von Neumann divergence, a particular Bregman divergence, to estimate the parameters of the structural equation model. Our approach is tested in a simulation study and shows significant advantages in estimating the model parameters in contaminated data sets and seems to perform better than other well known robust inference approaches in structural equation models.

**References**- E. G. Baranoff, S. Papadopoulos and T. W. Seger. Capital and Risk Revisited: A Structural Equation Model Approach for Life Insurers.
*Journal of Risk and Insurance,*74:653–681, 2007. doi:10.1111/j.1539-6975.2007.00229.x - K. Bollen.
*Structural Equations with Latent Variables*. Wiley, New York, 1989. - I. S. Dhillon and J. A. Tropp. Matrix Nearness Problems with Bregman Divergences.
*SIAM Journal on Matrix Analysis and Applications*, 29:1120–1146, 2008. doi:10.1137/060649021 - F. Nielsen and S. Boltz. The Burbea–Rao and Bhattacharyya Centroids.
*IEEE transactions in information theory*, 57:5455–5466, 2011. doi:10.1109/TIT.2011.2159046 - B. C. Vemuri, M. Liu, S.-I. Amari and F. Nielsen. Total Bregman Divergence and Its Applications to DTI Analysis.
*IEEE Transactions on medical imaging*, 30:475–483, 2011. doi:10.1109/TMI.2010.2086464 - S. Verboven and M. Hubert. LIBRA: a MATLAB library for robust analysis,
*Chemometrics and intelligent laboratory systems*, 75:127–136, 2005. doi:10.1016/j.chemolab.2004.06.003 - K.-H. Yuan and P. M. Bentler. Structural equation modeling with robust covariances.
*Sociological Methodology*, 28:363–396, 1998. doi:10.1111/0081-1750.00052 - K.-H. Yuan and P. M. Bentler. Robust mean and covariance structure analysis,
*British Journal of Mathematical and Statistical Psychology*, 51:63–88, 1998. doi:10.1111/j.2044-8317.1998.tb00667.x - K.-H. Yuan, P. M. Bentler and W. Chan. Structural Equation Modeling with Heavy Tailed Distributions,
*Psychometrika*, 69:421–436, 2004. doi:10.1007/BF02295644 - X. Zhong and K.-H. Yuan. Bias and Efficiency in Structural Equation Modeling: Maximum Likelihood Versus Robust Methods.
*Multivariate Behavioral Research,*46:229–265, 2011. doi:10.1080/00273171.2011.558736 - LIBRA: a Matlab Library for Robust Analysis. http://wis.kuleuven.be/stat/robust/LIBRA/LIBRA-home

#### Keywords

Total Bregman Distance, Structural Equation /model

DOI: http://dx.doi.org/10.21914/anziamj.v54i0.6306

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