On the power of the Kolmogorov test to detect the trend of a Brownian bridge with applications to a change-point problem in regression models

Given a Brownian bridge B0 with trend g:[0,1]-->[0,[infinity]), Y(z)=g(z)+B0(z),z[set membership, variant][0,1],we are interested in testing H0:g[reverse not equivalent]0 against the alternative K:g>0. For this test problem we study weighted Kolmogorov testswhere c>0 is a suitable constant and w:[0,1]-->[0,[infinity]) is a weight function. To do such an investigation a recent result of the authors on a boundary crossing probability of the Brownian bridge is useful. In case the trend is large enough we show an optimality property for weighted Kolmogorov tests. Furthermore, an additional property for weighted Kolmogorov tests is shown which is useful to find the more favourable weight for specific test problems. Finally, we transfer our results to the change-point problem whether a regression function is or is not constant during a certain period.