# 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?

• Please don't crosspost at M.SE and MO. – Michael Greinecker Jul 26 '12 at 15:26
• arxiv.org/abs/math/0209021 – Steve Huntsman Jul 26 '12 at 15:27
• @Michael: May I ask why? @Steve: I am aware of that article. However, as I have already admitted in the question, I am not very well equipped to understand all the technical finesse of measure theory. This is why I was hoping for an answer which relates more directly to the difference between two densities (or CDFs or CCDFs) at the tail... – miladydesummer Jul 26 '12 at 15:37
• See the discussion here: tea.mathoverflow.net/discussion/1181/… The main reason is that effort shouldn't be duplicated. – Michael Greinecker Jul 26 '12 at 16:58
• The "total variation distance", from my point of view, is the norm of the difference of two measures (which may or may not be probability measures). In the case of $TV(\mu_X,\mu_Y;\tau)$, that is the norm of the difference of the restrictions of $\mu_X$ and $\mu_Y$ to $[\tau,\infty)$. Those restrictions, of course, are not probability measures. – Robert Israel Aug 24 '12 at 0:28

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