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Is there a sensible notion of empirical cdf in several dimensions? If not, what other meaningful ways of saying "this collection of points in $\mathbb{R}^d$ looks like it could be iid from this multivariate distribution" are there?

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As long as I know there is an extension of the co-occurance matrix denoted by $$C(i,j), i=1,...,N-1, j=0,...,M-1$$ It is used in the context of image processing and essentially the extension of the histogram to multi dimensions. There are several papers who use multivariate co-occurance matrix in order to state the multivariate emprical density function. As the density can be obtained uniquely from the distribution function, one can say that this collection of $\mathbb{R}^d$ looks like it could be iid from this multivariate distribution.

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