exact distribution of change-point mle for a normal sequence

The article focuses on computing the exact distribution of the change-point maximum likelihood estimate (mle) in the scenario where the mean of an independent normal random process changes, at an unknown time, and the change magnitude and the variance parameter are known. then we use the resulting distribution as an approximation for the change-point estimate distribution when the amount of the change in mean is unknown, and evaluate its efficiency through simulation studies. simulations show that the exact distribution outperforms the asymptotic distribution. notably, even in the absence of a change, the exact distribution maintains its efficiency, a feature not shared by the asymptotic distribution. finally, the developed methodology is applied to construct confidence sets for the changepoint. this suggests that the proposed approach can provide reliable confidence sets that capture the true change-point with a good level of confidence, even when the magnitude of the change in the mean is unknown.