The asymptotic covariance matrix of maximum-likelihood estimates in factor analysis: the case of nearly singular matrix of estimates of unique variances

AbstractThis paper is concerned with the asymptotic covariance matrix (ACM) of maximum-likelihood estimates (MLEs) of factor loadings and unique variances when one element of MLEs of unique variances is nearly zero, i.e., the matrix of MLEs of unique variances is nearly singular. In this situation, standard formulas break down. We give explicit formulas for the ACM of MLEs of factor loadings and unique variances that could be used even when an element of MLEs of unique variances is very close to zero. We also discuss an alternative approach using the augmented information matrix under a nearly singular matrix of MLEs of unique variances and derive the partial derivatives of the alternative constraint functions with respect to the elements of factor loadings and unique variances