Maximum likelihood mean and covariance matrix estimation constrained to general positive semi-definiteness

Abstract

Maximum likelihood estimation of a mean and a covariance matrix whose structure is constrained only to general positive semi-definiteness is treated in this paper. Necessary and sufficient conditions for the local optimality of mean and covariance matrix estimates are given. Observations are assumed to be independent. When the observations are also assumed to be identically distributed, the optimality conditions are used to obtain the mean and covariance matrix solutions in closed form. For the nonidentically distributed observation case, a general numerical technique which integrates scoring and Newton's iterations to solve the optimality condition equations is presented, and convergence performance is examined.

Keywords:

• optimality conditions
• scoring
• Newton's method
• EM algorithm