Estimating a covariance matrix from incomplete realizations of a random vector

It is desired to estimate the mean and the covariance matrix of a Gaussian random vector from a set of independent realizations, with the complication that not every component of each realization of the random vector is observed. Subject to some restrictions, the authors obtained an exact, noniterative solution for the maximum likelihood (ML) estimates of the mean and the covariance matrix. The ML estimate of the covariance matrix that is obtained from the set of incomplete realizations is guaranteed to be positive definite, in contrast to ad hoc approaches based on averaging products of components from the same realization. The key to obtaining the ML estimates is a tractable expression for the likelihood function in terms of the Cholesky factors of the inverse covariance matrix. With this formulation, the ML estimates are found by fitting regression operators to appropriate subsets of the data. The Cholesky formulation also leads to a simple calculation by Cramer-Rao bounds