Dear experts,

I need to quantify the agreement between simulated and observed 1D and 2D histograms in order to select the model distribution (1D or 2D) that best describe the observed distribution. For the 1D histograms I know that I can obtain an estimate of the probability that the two 1D distributions are obtained from the same parent distribution by applying the 2 sample KS test. But this does not look true for the 2D distributions, as suggested in the article https://asaip.psu.edu/Articles/beware-the-kolmogorov-smirnov-test (basically because there are infinite ways do sort a 2D histogram). Eric Feigelson explained me that the 2 sample KS statistics gives a reasonable estimate of the “distance” between the two 2D sample but it cannot be used to give the probability that the samples are obtained from the same parent population without bootstrap resampling to find its distribution for your datasets.

Now, since I think that I can safely use the 2KS test to compare 1D distributions and to give the probability that those are obtained from the same parent distribution, can I use the 2KS statistics to compare my 2D distributions and discriminate the model that best explains the observations just by selecting the model (out of 4) that shows the smallest “distance” to the data (since I cannot use the probability here)?

Do you have any other statistical approach to suggest, both for 1D and 2D distributions, that I can use instead of the 2KS test and statistics?

Examples of 1D and 2D distributions are attached. Red points or line are observations, grey histograms are simulations. Each panel represent a model and I have to decide the one that best explains the observations.

Thank you very much,

Marco