Accessing the Power of Tests Based on Set-Indexed Partial Sums of Multivariate Regression Residuals

The intention of the present paper is to establish an approximation method to the limiting power functions of tests conducted based on Kolmogorov-Smirnov and Cramér-von Mises functionals of set-indexed partial sums of multivariate regression residuals. The limiting powers appear as vectorial boundary crossing probabilities. Their upper and lower bounds are derived by extending some existing results for shifted univariate Gaussian process documented in the literatures. The application of multivariate Cameron-Martin translation formula on the space of high dimensional set-indexed continuous functions is demonstrated. The rate of decay of the power function to a presigned value is also studied. Our consideration is mainly for the trend plus signal model including multivariate set-indexed Brownian sheet and pillow. The simulation shows that the approach is useful for analyzing the performance of the test.