Lisa Madsen and David Ruppert
_{Department of Statistics
Oregon State University
44 Kidder Hall, Corvallis OR 97331
E-mail: lmadsen@stat.orst.edu}

Abstract
Suppose X(s) and e(s) are stationary spatially autocorrelated Gaussian processes and Y(s) = ß0 +ß1X(s) + e(s) for any location s. Our problem is to estimate the ß's, particularly ß1, when X and Y are not necessarily observed in the same location. This situation may arise when the data are recorded by different agencies or when there are missing data values. A natural but naive approach is to predict ("krige") the missing X's at the locations Y is observed, and then use least squares to estimate ("regress") the ß's as if these X's were actually observed. This krige-and-regress estimator is consistent, even when the spatial covariance parameters are estimated. If we use it as a starting value for a Newton-Raphson maximization of the likelihood, the resulting maximum likelihood estimator is asymptotically effcient. We can then use an information-based variance estimator for inference.

Keywords: Spatial regression, Maximum likelihood, Misaligned data

In: McRoberts, R. et al. (eds).  Proceedings of the joint meeting of The 6th International Symposium On Spatial Accuracy Assessment In Natural Resources and Environmental Sciences and The 15th Annual Conference of The International Environmetrics Society, June 28 – July 1 2004, Portland, Maine, USA.