Estimating variance parameters from multivariate normal variables subject to limit of detection: MLE, REML, or Bayesian approaches?

Likelihood-based approaches, which naturally incorporate left censoring due to limit of detection, are commonly utilized to analyze censored multivariate normal data. However, the maximum likelihood estimator (MLE) typically underestimates variance parameters. The restricted maximum likelihood estimator (REML), which corrects the underestimation of variance parameters, cannot be easily extended to analyze censored multivariate normal data. In the light of the connection between the REML and a Bayesian approach discovered in 1974 by Dr Harville, this paper describes a Bayesian approach to censored multivariate normal data. This Bayesian approach is justified through its link to the REML via Laplace's approximation and its performance is evaluated through a simulation study. We consider the Bayesian approach as a valuable alternative because it yields less biased variance parameter estimates than the MLE, and because a solid REML is technically difficult when data are left censored.