On the supremum of a Brownian bridge standardized by its maximizing point with applications to statistics

Let T be the point, where a reflected Brownian bridge attains its maximal value M. We determine the density and the distribution function of R≔M∕T(1−T). Its counterpart Rn for tied-down sums pertaining to i.i.d. random variables converges in distribution to R. This enables the construction of a change-point test. Our new test performs significantly better than the well-known maximum-type test statistics.