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…
Best Answer
I gave you this same answer over at MathOverflow. Looking at your response to Michael Chernick, you probably do want to consult Dudley, as the Prohorov metric and its follow-up in Proposition 11.3.2 refer directly to metrics on random variables, which could be defined for the tail only as you request.
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.