Asymptotic Robust Inferences in the Analysis of Mean and Covariance Structures
Structural equation models are widely used in economic, social and behavioral studies to analyze linear interrelationships among variables, some of which may be unobservable or subject to measurement error. Alternative estimation methods that exploit different distributional assumptions are now available. The present paper deals with issues of asymptotic statistical inferences, such as the evaluation of standard errors of estimates and chi--square goodness--of--fit statistics, in the general context of mean and covariance structures. The emphasis is on drawing correct statistical inferences regardless of the distribution of the data and the method of estimation employed. A (distribution--free) consistent estimate of $\Gamma$, the matrix of asymptotic variances of the vector of sample second--order moments, will be used to compute robust standard errors and a robust chi--square goodness--of--fit squares. Simple modifications of the usual estimate of $\Gamma$ will also permit correct inferences in the case of multi-- stage complex samples. We will also discuss the conditions under which, regardless of the distribution of the data, one can rely on the usual (non--robust) inferential statistics. Finally, a multivariate regression model with errors--in--variables will be used to illustrate, by means of simulated data, various theoretical aspects of the paper.
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