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.

Keywords

• heteroscedastic linear regression model, least squares residual, partial sums process, Brownian sheet, asymptotic model-check.