Exact distribution of change-point MLE for a Multivariate normal sequence

This paper presents the derivation of an expression for computing the exact distribution of the change-point maximum likelihood estimate (MLE) in the context of a mean shift within a sequence of time-ordered independent multivariate normal random vectors. The study assumes knowledge of nuisance parameters, including the covariance matrix and the magnitude of the mean change. The derived distribution is then utilized as an approximation for the change-point estimate distribution when the magnitude of the mean change is unknown. Its efficiency is evaluated through simulation studies, revealing 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. To demonstrate the practical application of the developed methodology, the monthly averages of water discharges from the Nacetinsky creek in Germany are analyzed, and a comparison with the analysis conducted using the asymptotic distribution is presented.