Asymptotic approach to model testing for heteroscedastic spatial regression with independent observations

We derive a functional central limit theorem for heteroscedastic spatial regressions by applying the generalized version of Prohorov’s theorem. By our technique we get the limit process which is expressed as a function of a centered set-indexed Gaussian process including the standard set-indexed Brownian sheet as a special case. The result can be used to approximate the distributions of a type of Kolmogorov-Smirnov (KS) and Cramér-von Mises (CvM) functionals of the set-indexed partial sums (Cumulative Sum) processes of the least squares residuals which are useful for testing the adequateness of an assumed regression model. A simulation study is performed to investigate the finite sample sizes behavior of the tests. It is shown by simulation that among the two tests, the CvM test tends to have better power than KS test for testing first-order model against nonparametric or parametric alternative. An application of the established method in real data is also discussed.