The rate of decay of a Kolmogorov-Smirnov functional of the multivariate Slepian field with a deterministic trend

In this work we aim to give an investigation to the probability of a Kolmogorov-Smirnov (K-S) functional of the multi-variate Slepian field with a trend. This type of problem which is called in the literatures as the boundary crossing probability has been appeared in many disciplines, such as in physics, engineering, finance mathematics and statistics and also in industry. As a typical problem in statistic this kind of probability has been obtained as the limit of the power function of a test for the validity of the vector mean of a multivariate linear regression based on the K-S functional of the moving sums of the vector of least squares residuals. In this paper upper and lower bounds for the rate of decay of the power function to any pre signed value of the size of test are derived by applying the well-known Wenbo-Kuelb and the Cameron-Martin density formula of shifted multivariate centered Gaussian process. We also derive the upper and lower bounds for the rate of decay of the power function of a most powerful test which is involving the multivariate Slepian field and comparing both approaches by Monte Carlo simulation.