When sustained changes due to special causes of variation are present, tests and control charts can be used to detect them, and change-point estimators can be applied to approximate their location. When dealing with normal observations, classical maximum likelihood estimation of a change-point does not consider prior knowledge about the change-point. Also, if confidence intervals are required, the corresponding distribution of the likelihood ratio is cumbersome and difficult to implement. To assess these issues, Bayesian approaches implement prior distributions to the unknown parameters. However, prior distributions cannot always be assumed for all parameters. A middle ground can be found by using hierarchical models, where only a portion of the parameters have a prior distribution, and the rest is estimated from the sample through the use of hierarchical maximum likelihood estimation. In this paper, by assuming prior knowledge of the distribution of the change-point, a hierarchical change point model is derived and the corresponding estimation and confidence intervals are obtained. When prior knowledge of the change-point exists, this approach might provide less biased estimations with smaller standard error. Additionally, by using the posterior distribution of the change-point, exact confident intervals can be derived.
