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 SIGMA-0* that is the sum of a matrix SIGMA(gamma-0) that satisfies a specified model and a perturbation matrix, E, such that SIGMA-0* = SIGMA(GAMMA-0) + E. The perturbation matrix is chosen in such a manner that a class of discrepancy functions F(SIGMA-0*, SIGMA(gamma-0)), which includes normal theory maximum likelihood as a special case, has the prespecified parameter value gamma-0 as minimizer and a prespecified minimum-delta. 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.