  We are concerned with the problem of detecting a single change point in the
model parameters of time series data generated from an exponential family. In
contrast to the existing literature, we allow that the true location of the
change point is itself random, possibly depending on the data. Under the
alternative, we study the case when the size of the change point converges to
zero while the sample size goes to infinity. Moreover, we concentrate on change
points in the "middle of the data", i.e., we assume that the change point
fraction (the location of the change point relative to the sample size)
converges weakly to a random variable $\lambda^*$ which takes its values almost
surely in a closed subset of $(0,1).$ We show that the known statistical
results from the literature also transfer to this setting. We substantiate our
theoretical results with a simulation study.
