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I have a basic question about copula. I am not an expert in statistics myself but use statistics for modelling and data analysis a lot.

I have read in multiple sources and also in Wikipedia that:

This implies that the copula is unique if the marginals F_i are continuous.

I cannot understand why it is said that copula is unique if marginals are continuous.

If copula is unique, why do we consider different formulations (e.g., Archimedean copula) for them and test to see which one results in better fit? What is the part that I am missing?

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If I recall correctly, the uniqueness under continuity is a corollary of Sklar's theorem, which itself guarantees existence for continuous and discrete distributions. See Nelsen 2009.

A given collection of marginal distributions can be mapped to different joint distributions. For a given joint distribution the copula specifies how the marginals map to it, which is unique for continuous distributions. Different copulas applied to a given set of continuous distributions will give a different joint distribution.

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  • $\begingroup$ Continous margins guarantee no gaps or jumps in your function. Hence, we can apply a unique copula. However, we do not know which copula fits the data. Hence, we try to select the best-fit one (unique in this case). $\endgroup$
    – Dr. Statistics
    Commented Feb 19, 2023 at 5:56

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