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

Abstract

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 α 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.

Simultaneous Model Change Detection in Multivariate Linear Regression With Application to Indonesian Economic Growth Data

“…Although the limit process has a complex structure, nevertheless it is geometrically interpretable as a projection of the standard multivariate Brownian motion on its kernel space. This simple structure gives advantage in analyzing the power of the test when the assumed model is not true (see Somayasa [9]).…”