Ridge-type covariance and precision matrix estimators of the multivariate normal distribution

We consider ridge-type estimation of the multivariate normal distribution’s covariance matrix and its inverse, the precision matrix. While several ridge-type covariance and precision matrix estimators have been presented in the literature, their respective inverses are often not considered as precision and covariance matrix estimators even though their estimands are one-to-one related through the matrix inverse. We study which estimator is to be preferred in what case. Hereto we compare the ridge-type covariance matrix estimators and their properties to that of the inverse of the ridge-type precision matrix estimators, and vice versa. The comparison, in which we take all ridge-type estimators along, is limited to a specific case that is illustrative of the difference between the covariance and precision matrix estimators. The comparison addresses the estimators’ estimating equation, analytic expression, analytic properties like positive definiteness and penalization limit, mean squared error, consistency, Bayesian formulation, and their loss and potential for marginal and partial correlation screening.