Estimating the power of lack-of-fit test for the mean of multivariate spatial regression

Abstract

The purpose of this paper is to establish an estimation of a boundary crossing probability composed by a multivariate centered Gaussian processes and a vector of deterministic trends. Such probability corresponds in statistics to the power function of an asymptotic lack-of-fit test conducted based on the Kolmogorov functional of set-indexed partial (cumulative) sums process of the least squares residuals of multivariate spatial regression. Since the analytical computation of the probability is impossible, we investigate its upper and lower bounds by applying some methods relied on the multivariate Cameron-Martin translation formula on the space of high dimensional set-indexed continuous functions. Our consideration is mainly for the multivariate set-indexed Brownian pillow. The results are shown not only useful for analyzing the behavior of the test, but also worth for the abstraction and generalization of some existing results toward univariate Gaussian process studied in many literatures.