Stochastic Model Validation

Hi, I am working on a model that takes in a stochastic time series (turbulent velocity) as input and produces n time series outputs. Since the inputs have a stochastic component, every ensemble (or every run of the model) produces slightly varying outputs. Now I want to validate my model by comparing the output quantities obtained using the same input but from a different model which is assumed to be accurate.

Since the models have stochastic input, a simple time series (visual) comparison is hardly full proof. I could extend the comparison in the spectral space, to compare PSDs of the outputs. However, I would like to see if I can show that the two models have the same higher order statistics. My question is as follows:

How do I generate the PDF of the output quantities and estimate the first few moments (mean, std, skewness, flatness) if they are represented as time series. Do I generate 100 ensembles and then create a pdf at each point in the time history? Furthermore, how can I calculate the cross-covariances of two quantities that I feel are "mechanistically" related? If I can show that the two models have similar statistics for all QoIs, then can I conclude that the results between the two models are consistent?