I have a bivariate dataset on $[0,1]^2$ in which I am interested in fitting a joint distribution. I fit a Gaussian copula but am unsure how to judge if it's a good fit. I tried transforming my data using the fitted cdf to see if the transformed data would be uniform on $[0,1]$. It turned out to be far from uniform.
I then compared it to the transform from the empirical cdf which gave a very similar plot.
Since the empirical cdf was fitted using about a mio datapoints I'd assume it's a pretty good approximation of the true joint CDF. Am I wrong to expect the transformed data to be uniform or could there be something else going wrong?