What might be some appropriate purposes for problem statements and purpose statements within doctoral research?
I assume from your previous questions that you are asking about problem statements and purpose statements here. I have changed your question accordingly.
The problem statement is meant mainly to lay out why the research that you are doing for your dissertation is important. One purpose for this is so that you can communicate the importance of the study to your committee and to other readers. A second purpose is to force you as the researcher to be very clear on why it is that you are undertaking this study. If you cannot state this clearly, you may need to find another topic.
The purpose statement is meant to lay out how you will study the problem that you have raised. As with the problem statement, a major reason to do this is to force yourself to be clear on what it is you are doing. You need to be able to identify a way of investigating the problem that will be neither too broad nor too narrow for a dissertation.
The major purposes of these statements, then, is to inform your readers of the importance and scope of the study and to force yourself to be perfectly clear on those issues as well.