Approximate Likelihoods for Spatial Processes

ABSTRACT
Many applications of spatial statistics involve evaluating a likelihood over samples of several hundred data locations. If the underlying field is Gaussian with some spatial covariance structure, this evaluation involves calculating the inverse and determinant of the covariance matrix. Although this is feasible for up to about 100 observations, it is often troublesome for sample sizes larger than 100. To take advantage of the benefits of maximum likelihood estimates for large arrays of data, it is necessary to establish efficient approximations to the likelihood. We consider several such approximations based on grouping the observations into clusters and building an estimating function by accounting for variability both between and within groups. This way, the estimation becomes practical for considerably larger data sets. In this talk we present the proposed alternatives to the likelihood function, and an analysis of the asymptotic efficiency of the estimators yielded by them. The theoretical method applies to any kind of spatial process, but an analogous time series model is used for illustration and explicit computation. In this context, since the standard Fisher information techniques of calculating the asymptotic variance of the alternative estimators would not lead to correct conjectures, we employ a method based on the ``information sandwich'' technique and a Corollary to the Martingale Central Limit Theorem (application to quadratic forms of independent normal random variables).  Furthermore, we illustrate the asymptotic behavior of the alternative parameters in the spatial setting with results from a simulation study.

Refreshments will be served at 3:45 p.m. in Room 008 of the Statistics Building.

College of Natural Sciences