# 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

(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?

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.

• I will check out the book as soon as I can get my hands on it. However, looking at the Google preview of Chapter 11, I am not sure I will be able to draw very much from it. As I've said in the question and again in the comments, 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 15:41
• Unfortunately, metrics on random variables are a bit complicated. The second metric derived in 11.3.2 is the sup of the difference in two random variables, so it's probably what you want, but without more of a background in theoretical probability you may not be able to use it as you want. Also, Michael's comment means you should try to pick the right forum and post there. In this case, M.SE is probably more appropriate because your question is not research level, per the MO FAQ. – user5794 Jul 26 '12 at 16:14