Finding shifts in weighted data

Hello everyone,

I'm working with large data sets in which every element has its own weight. In this sample different changes (shifts, scales, shifts & scales, even distribution changes) are possible. I'm trying to solve the problem of locating and determining shifts in it.

By the moment I've thought of some method which checks whether the point between two observations is a shift point or not. This method uses Mann-Whitney U-test but there is an obvious weakness of this method: as I understand, as H0 we take "two observed samples have equal distributions", while I can not guarantee that. Also, I don't understand how to use weights in this test.

Also, I've thought of some change point detection methods, such as CUSUM. Unfortunately, pure CUSUM method doesn't work properly with my sets. Also, I don't understand how the weights should be applied there.

The question therefore is if there are any methods of locating shift points in such a set which take weights into account and do not demand equality of probabilities.

Any help is highly appreciated!

Thanks in advance!