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Two finite-sampled continuous random variables $X$ and $Y$ are transformed so that they are uniformly distributed, $U$ and $V$. With these as marginals, the empirical copula of $X$ and $Y$, denoted $C(u,v)$, can be estimated. The estimation approach being asked here is by discretization via histogram binning.

Since alot of copula theory sticks to just that, theory (integrals, therefore, non-discrete), the discretization implementation is hardly covered. Could someone provide a source from a copula textbook that explains this formula found in the numpy histogram2d documentation:

probability density function at the bin = bin_count / sample_count / bin_area.

By 'at the bin' does this mean there is a distribution formed for each bin? and why the triple fraction? I'm trying to make sense of it for the following:

copula_density = np.histogram2d(X_unif, Y_unif, bins, density=True)[0]
copula_freq = np.histogram2d(X_unif, Y_unif, bins, density=False)[0]
bin_area = (copula_freq / copula_density / len(X_unif))[0, 0]

since the last line looks like it backs out the bin_area from the same equation shown earlier, being a triple fraction also. I also don't know the importance of bin_area or what application it would be required for afterwards.

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  • $\begingroup$ Probability integral transform (PIT) is used for transforming continuous random variables to uniform, and this is done when working with copulas. I wonder what the justification for other types of transformations is. $\endgroup$ Commented Aug 28, 2020 at 17:12
  • $\begingroup$ are you saying the binning approach above is different than integral transform (inverse CDF)? $\endgroup$
    – develarist
    Commented Aug 28, 2020 at 17:17
  • $\begingroup$ I don't know. If it were the same, why a different name? $\endgroup$ Commented Aug 28, 2020 at 17:32
  • $\begingroup$ By definition, the density is a value per unit area. Evidently, then, this estimate of the PDF is obtained by estimating probabilities as fractions of the sample count. $\endgroup$
    – whuber
    Commented Aug 28, 2020 at 18:24
  • $\begingroup$ "area" to me is length $\times$ width. how does this concept possibly match up with bin_area = copula_freq / copula_density / len(X_unif)? $\endgroup$
    – develarist
    Commented Aug 29, 2020 at 12:59

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