Local Polynomial Estimation for a Smooth Spatial Random Process with a Stochastic Trend and a Stationary Noise

We consider the problem of estimating the parameters of the covariance function of a stationary spatial random process. In spatial statistics, there are widely used parametric forms for the covariance functions. We develop a method for estimating the parameters of the covariance function that is based on a regression approach. Our method utilizes pairs of observations whose distances are closest to a value h > 0 which is chosen in a way that the estimated correlation at distance h is a predetermined value. We demonstrate the effectiveness of our procedure by simulation studies and an application to a water pH data set. We also show that under a mixing condition on the random field, the proposed estimator is consistent for standard one parameter models for stationary correlation functions. Next, we model the observed spatial random process as a sum of the stochastic trend, a short term stationary error and a measurement error. Then we estimate the stochastic trend nonparametrically using a local polynomial method and obtain an expression for the asymptotic mean squared error of the trend estimate. Our asymptotic analysis shows that the asymptotic mean squared error for the stochastic trend model is of the same order of magnitude as that for the deterministic trend model. We also propose a data dependent selection procedure for the smoothing parameter (bandwidth associated with the local polynomial method). We provide simulation studies and applications to real data.