I need to compare whether two distributions are similar when the values are scaled by the mean of each of the distribution. One limitation of ks-test as per http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm is "if location, scale, and shape parameters are estimated from the data, the critical region of the K-S test is no longer valid."
Consider for example:
Data1 consist of 10000 numbers from uniform random distribution [0,1] with mean 0.5
Data2 consist of 10000 numbers from uniform random distribution [0,10] with mean 5.001
If I compare Data1 with Data2/10 then ks-test gives that both the distribution are same; while comparing Data1/0.5 with Data2/5.001 gives that the distribution are different. Is there a way to check the similarity between the distributions in such cases?
Edit: As the answer suggests I can use ks-test where the p-value is determined via permutation.
My additional difficulty is that the data-points are integers:
Data1 consist of 10000 integers from uniform random distribution [0,10] with mean 5
Data2 consist of 10000 integers from uniform random distribution [0,100] with mean 50.001
Is there a test to compare whether
Data2 are similar apart from the scale? Further, I do not know the actual scale and I am determining it from the data.
These examples are just a proxy for my actual data, which are two experiment where 10000 people rated a movie on a scale [0,10], while in other case 10000 different people rated the same movie on a scale [0,100]. I want to check apart from the scale can one say that whether the distributions are same or not.