I know that both the ROC curve and the PR curve can be used to evaluate the performance of a binary classification prediction model, and PR curve is preferred in the case of imbalanced class distribution. In this regard, I've got a question but couldn't figure out the answer by myself nor could find any in the textbook or other sources.
Here is my question:
Let's assume that the test set is fixed and we have two binary classification models, A and B. If the model A has a larger AUC in terms of ROC curve than the model B, does it necessarily mean that the model A has a larger AUC in terms of PR curve as well? In other words, does the AUC of ROC curve has a monotonic relationship with the AUC of PR curve given a fixed test set? If it's not the case, I would greatly appreciate if some counterexamples can be shown.
Thank you in advance!