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Abstract
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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.
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