Asymptotics of a boundary crossing probability of a Brownian bridge with general trend

Let us consider a signal-plus-noise model yh(z) + B-0(z), z epsilon [0, 1], where gamma > 0, h : [0, 1] --> R, and B-0 is a Brownian bridge. We establish the asymptotics for the boundary crossing probability of the weighted signal-plus-noise model for gamma --> infinity, that is P(sup(zepsilon(0,1)) w(z) (gammah(z) + B-0(z)) > c), for gamma --> infinity, (1) where w : [0, 1] --> [0, infinity) is a weight function and c > 0 is arbitrary. By the large deviation principle one gets a result with a constant which is the solution of a minimizing problem. In this paper we show an asymptotic expansion that is stronger than large deviation. As byproduct of our result we obtain the solution of the minimizing problem occurring in the large deviation expression. It is worth mentioning that the probability considered in (1) appears as power of the weighted Kolmogorov test applied to the test problem H-o : h equivalent to 0 against the altemative K : h > 0 in the signal-plus-noise model.