Abstract
A method is presented for constructing a covariance matrix Σ*0 that is the sum of a matrix Σ(γ0) 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 functionsF(Σ*0, Σ(γ0)), which includes normal theory maximum likelihood as a special case, has the prespecified parameter value γ0 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.
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We are grateful to Alexander Shapiro for suggesting the proof of the solution in section 2.
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Cudeck, R., Browne, M.W. Constructing a covariance matrix that yields a specified minimizer and a specified minimum discrepancy function value. Psychometrika 57, 357–369 (1992). https://doi.org/10.1007/BF02295424
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DOI: https://doi.org/10.1007/BF02295424