A nonparametric covariance estimator for spatial models

Torgny Lindström

Abstract:: The covariances in spatial models are estimated by linear smoothing of products of residuals. In the model no parametric assumptions are made about the mean function or the spatial dependence. Both are assumed to be smooth. Smoothing is based on local polynomials, though any linear smoother is possible to use. Expressions for the mean and the covariance of this estimator are developed and a version that corrects for bias is proposed. Note that the covariance estimates generated by this method are not guaranteed to be positive definite, though proper covariance function estimates can be generated by known methods. The advantage with the covariance estimation described here is that the procedure might allow for testing of stationarity prior to the fitting of a stationary covariance function. Simulation studies are performed to observe the estimator both for a stationary and isotropic, and a heteroscedastic model. We show good agreement between numerical and theoretical results and also numerically explore the bias introduced by the different smoothers used in the estimation procedure.

Key words:: Bivariate local polynomial regression, heteroscedastic data, local bandwidths, nonstationary model, smoother matrix