Kriging Using James-Stein Estimators which Contract to Subspaces



In universal kriging we have a response and a vector of explanatory variables, observed at a number of sites in the plain. Our objective is to fit a linear model to predict future responses at given additional sites in the plain, taking into account the spatial autocorrelation among the error terms in the sample AND their autocorrelation with the error terms for the future responses. This can be formulated as a problem in weighted least squares (WLS) regression, and Kriging is the optimal unbiased linear estimator (named after the South African mining engineer D G Krige).

Since this is essentially a problem in WLS regression, it is natural to think of using the James-Stein (JS) estimator. This has been done by several authors, but there do not appear to be any convincing applications. This is probably because they contracted ˆ β to the origin; and if we think β = 0, why are we doing the regression?

Here, we adapt a JS method, previously developed for contracting a multivariate normal random vector to a subspace, to contracting ˆ β in the Kriging context. We apply it to three real data sets, and obtain substantial improvement.