Model-Check Based on Residual Partial Sums Process of Heteroscedastic spatial Linear Regression Models

Abstract

It is common in practice to evaluate the correctness of an assumed linear regression
model by conducting a model-check method in which the residuals of the observations are
investigated. In the asymptotic context instead of observing the vector of the residuals directly,
one investigates the partial sums of the observations. In this paper we derive a functional central
limit theorem for a sequence of residual partial sums processes when the observations come
from heteroscedastic spatial linear regression models. Under a mild condition it is shown that
the limit process is a function of Brownian sheet. Several examples of the limit processes are
also discussed. The limit theorem is then applied in establishing an asymptotically Kolmogorov
type test concerning the adequacy of the fitted model. The critical regions of the test for finite
sample sizes are constructed by Monte Carlo simulation.