Likelihoods for Spatial Processes
University of North Carolina-Chapel Hill
Friday, 07 February 2003
201 Glover Building
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