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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?

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  • $\begingroup$ Can you find any example where the CDF transforms a multivariate random variable into a uniform variable? $\endgroup$
    – whuber ♦
    Commented Nov 13, 2023 at 2:53

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I cannot add a comment, so I'm writing it in an answer.

Am I wrong to expect the transformed data to be uniform or could there be something else going wrong?

It is the marginal CDF, which is assumed continuous, of each component of multivariate random variable that is uniform on the interval [0, 1]. But the CDF transform of a multivariate random variable is not uniform on [0, 1[.

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  • $\begingroup$ Doing some numerical experiments this now became very clear. Is there a good theoretical reason for why it doesnt hold in the multivariate case? $\endgroup$
    – Bpe
    Commented Nov 14, 2023 at 10:01
  • $\begingroup$ It is not held for the CDF transform, that is because not all transformation functions preserve the independence between the two variables. $\endgroup$
    – Maryam
    Commented Jan 3 at 16:55

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