Some asymptotic results for the residual maximum likelihood estimation of the parameters of a spatial process

Abstract

Zhang and Stein (1993) gave a kernel approximation to the order m(m≥1) universal kriging predictor in two dimensions under an order mintrinsic random function model. The two parameters involved in the model are the signal variance and the noise variance. To employ the likelihood method for parameter estimation, we adopt a residual maximum likelihood procedure based on the contrasts of the observations. To explore various asymptotic properities of the residual maximum likelihood estimates,we introduce a continuous stochastic model on the sphere and argue that the limiting behavior in the continuous setting is related to the discrete problem of interest. We derive the asymptotic distribution of the maximum likelihood estimate of the signal variance as the noise variance tends to zero. Using several numerical studies,we show that the approximation of the Fisher information for the signal variance in the continuous case to that in the discrete case works fairly well.