Constructing a Covariance Matrix that Yields a Specified Minimizer and a Specified Minimum Discrepancy Function Value

A method is presented for constructing a covariance matrix Σ* 0 that is the sum of a matrix Σ(γ 0 ) that satisfies a specified model and a perturbation matrix, E , such that Σ* 0 =Σ(γ0) + E . The perturbation matrix is chosen in such a manner that a class of discrepancy functions F (Σ* 0 , Σ(γ 0 )), which includes normal theory maximum likelihood as a special case, has the prespecified parameter value γ 0 as minimizer and a prespecified minimum δ A matrix constructed in this way seems particularly valuable for Monte Carlo experiments as the covariance matrix for a population in which the model does not hold exactly. This may be a more realistic conceptualization in many instances. An example is presented in which this procedure is employed to generate a covariance matrix among nonnormal, ordered categorical variables which is then used to study the performance of a factor analysis estimator.