A numerical procedure for finding the positive definite matrix closest to a patterned matrix

Patterned covariance matrices arise naturally from models in the physical, biological, psychological and social sciences. When the underlying data arises from a multivariate normal distribution, maximum likelihood estimates of the population covariance matrix can be obtained numerically, via an iterative procedure, or in some special cases, as closed form expressions. Without the assumption of normality we address the problem of obtaining an estimator that has the appropriate pattern and is close to the sample covariance matrix.