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(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. We are grateful to Alexander Shapiro for suggesting the proof of the solution in section 2.