Merge and chop in the computation for isotonic regressions

In this article the sufficient conditions for the merge and chop of domain partition sets during the computation for isotonic regressions are derived. The proposed merge-and-chop procedure yields exact isotonic regressions in finite steps when the orderings are of Type I or II defined in this paper. Simple tree orderings, umbrella orderings and simple orderings are of Type I or II. The paper also provides an example showing that the procedure works beyond Types I and II orderings.