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We are concerned with the problem of detecting a single change point in themodel parameters of time series data generated from an exponential family. Incontrast to the existing literature, we allow that the true location of thechange point is itself random, possibly depending on the data. Under thealternative, we study the case when the size of the change point converges tozero while the sample size goes to infinity. Moreover, we concentrate on changepoints in the "middle of the data", i.e., we assume that the change pointfraction (the location of the change point relative to the sample size)converges weakly to a random variable $\lambda^*$ which takes its values almostsurely in a closed subset of $(0,1).$ We show that the known statisticalresults from the literature also transfer to this setting. We substantiate ourtheoretical results with a simulation study.

Parametric change point detection with random occurrence of the change point