Some Properties of Best Linear Unbiased Predictors and Related Predictors.

The variation of temperature or of pollutant concentrations over a geographic area are adequately represented by random fields. Given a real-valued random field zx,x an element of R-squared a basic problem is to interpolate Z over an area A from measurements taken at n stations x1,x2,...,xn, when the distribution of Z is only partially specified. This is the motivation of the present paper. It is shown that if the joint distributions are Gaussian the best linear unbiased predictor B.L.U.E. is among other properties admissible when used to predict Z at a single point, but in admissible in general when used to predict the values of the field at several points. A Stein-like predictor is produced which is uniformly better than the B.L.U.E. in the latter case. A nonlinear predictor, based on relaxing the unbiasedness condition on the B.L.U.E., is also proposed and shown to be in some cases preferable. Author