Re: More Difficult Follow Up Question on Corr Normal RVs


Brenneman wrote:
1) What measure do you use to quantify the extent to which they are
dependent, when you cannot use the correlation?

I realise you've dealt with the specific question, but I wanted to
answer this general question.

Dependence is a much more general concept than correlation.

Let me be a bit informal while I try to convey some basic thoughts.

For X and Y to be independent, you need the conditionals to be the same
as the marginals. There are uncountably infinite ways in which this may
be untrue. However, you could take some measure of dissimilarity of
f_{Y|X}(y|X=x) and f_Y(y), and integrate that over x, to get some
measure of the total extent of deviation from independence (though,
being a single number, it wouldn't tell you about the form of
dependence).

If you want to look at form of the general relationship between two
random variables, one approach is via copulas; in effect (and again,
informally), transform both marginals to uniformity, and look at how
the bivariate distribution differs from a bivariate uniform. This
serves to separate issues of the distributions of the margins from the
broader issue of the structure of the dependence between the variables.

I don't know if any of this helps.

Glen

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