Information loss on the mean for spatial processes when some values are missing

The loss of information on the mean due to the presence of missing values is discussed for a Gaussian univariate process on a rectangular lattice. The exact as well as the approximate formulae for this loss are given for general conditional autoregressive (CAR) and simultaneous autoregressive (SAR) processes. The formulae are evaluated for some low order CAR and SAR processes. The approximate formula is shown to give a good insight into how the loss varies over the different configurations of missing sites.