Fitting Covariance Structure Models With Arbitrary Observed Distributions: Strong Convergence of the Asymptotically Distribution-Free Estimator

The large-sample behavior of the asymptotically distribution-free estimator in covariance structure analysis is studied. It is shown that with increasing sample size and under suitable conditions, the estimator almost surely converges to the true parameter value. This strong convergence (i) represents numerical convergence with probability 1 of the resulting estimates of model parameters to their respective population values, as well as (ii) is stronger than the consistency and convergence in distribution of a parameter estimator, thus implying the latter two types of large-sample behavior. The demonstrated asymptotic convergence of the asymptotically distribution-free estimator for covariance structure models is illustrated on data.