measuring distance between probability measures only at the tail Is there any official (i.e., to be found in probability books) metric for the distance between two probability measures, defined only on a subset of their support?
Take, for example, the total variation distance: $$TV(\mu,\nu) = \sup_{A\in\mathcal{F}}|\mu(A)-\nu(A)|.$$
If $X$ and $Y$ are two real positive continuous random variables with densities $f_{X}$ and $f_{Y}$, then their total variation distance is, if I understand correctly: $$TV(\mu_{X},\mu_{Y}) = \int_{0}^{\infty} |f_{X}(z) - f_{Y}(z)|dz.$$
Would it make any sense to calculate a quantity, for $\tau > 0$, let's call it partial distance, like this:
$$PV(\mu_{X},\mu_{Y};\tau) = \int_{\tau}^{\infty} |f_{X}(z) - f_{Y}(z)|dz\;\;\;?$$
If this does not make any sense (sorry, I really cannot tell, as I am not that good with measure theory...), can anyone think of a measure that would make sense?
What I want to use this for is to compare the closeness of two PDFs (or other functions describing a distribution: CDF, CCDF...) $f_{X}(t)$, $f_{Y}(t)$ to a third one $f_{Z}(t)$. I know that both $f_{X}$ and $f_{Y}$ "eventually" ($t\to\infty$) converge to $f_{Z}$, but I would like to show that one of them gets closer, sooner than the other one...
EDIT: I guess the underlying question for measure-theory people is: do these distance metrics (like total variation, Kullback-Leibler etc) really need to span all the elements of the sample space of the probability measure and if so, why?
 A: You may want to check out Real Analysis and Probability by R. M. Dudley (2002, Cambridge University Press).  Chapters 9-11 discuss several metrics on probability measures and random variables (laws), and since restricting your support would be equivalent to some random variable on the measure, you should be able to use something like the metrics discussed in section 11.3 in particular.
A: Something like this is what I was looking for:
"A measure of discrimination between two residual life-time distributions and its applications" by Nader Ebrahimi and S.N.U.A. Kirmani from the Annals of the Institute of Statistical Mathematics, Volume 48, Number 2 (1996), 257-265
Can be downloaded from:
(official) Link
(maybe official?) http://www.ism.ac.jp/editsec/aism/pdf/048_2_0257.pdf
EDIT: I realize the proposal in the paper is not a metric (because not symmetric), but just to give an example of what I have in mind... Would a similar adaptation be possible for other measures of distance which are metrics?
