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International Journal of Mathematics and Mathematical Sciences
Volume 2016, Article ID 2601601, 16 pages
Research Article

Asymptotic Theory in Model Diagnostic for General Multivariate Spatial Regression

1Department of Mathematics, Haluoleo University, Kendari, Indonesia
2Department of Geological Engineering, Haluoleo University, Kendari, Indonesia

Received 25 March 2016; Revised 13 July 2016; Accepted 28 July 2016

Academic Editor: Andrei I. Volodin

Copyright © 2016 Wayan Somayasa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We establish an asymptotic approach for checking the appropriateness of an assumed multivariate spatial regression model by considering the set-indexed partial sums process of the least squares residuals of the vector of observations. In this work, we assume that the components of the observation, whose mean is generated by a certain basis, are correlated. By this reason we need more effort in deriving the results. To get the limit process we apply the multivariate analog of the well-known Prohorov’s theorem. To test the hypothesis we define tests which are given by Kolmogorov-Smirnov (KS) and Cramér-von Mises (CvM) functionals of the partial sums processes. The calibration of the probability distribution of the tests is conducted by proposing bootstrap resampling technique based on the residuals. We studied the finite sample size performance of the KS and CvM tests by simulation. The application of the proposed test procedure to real data is also discussed.