It is possible to go further, much more actually, on bounding sums of random variables (mentioned in the previous post). For instance, if everything has been defined, in that previous post, on distributions on , it is possible to extend bounds of distributions on
. Especially if we deal with quantiles. Everything we've seen remain valid. Consider for instance two
distributions.
Using the previous code, it is possible to compute bounds for the
quantiles of the sum of two Gaussian variates. And one has to remember that those bounds are sharp.
> Finv=function(u) qnorm(u,0,1) > Ginv=function(u) qnorm(u,0,1) > n=1000 > Qinf=Qsup=rep(NA,n-1) > for(i in 1:(n-1)){ + J=0:i + Qinf[i]=max(Finv(J/n)+Ginv((i-J)/n)) + J=(i-1):(n-1) + Qsup[i]=min(Finv((J+1)/n)+Ginv((i-1-J+n)/n)) + }
Actually, it is possible to compare here with two simple cases: the independent case, where the sum has a distribution, and the comonotonic case where the sum has a
distribution.
> lines(x,qnorm(x,sd=sqrt(2)),col="blue",lty=2) > lines(x,qnorm(x,sd=2),col="blue",lwd=2)
On the graph below, the comonotonic case (usually considered as the worst case scenario) is the plain blue line (with here an animation to illustrate the convergence of the numerical algorithm)
Below that (strong) blue line, then risks are sub-additive for the Value-at-Risk, i.e.
but above, risks are super-additive for the Value-at-RIsk. i.e.
(since for comonotonic variates, the quantile of the sum is the sum of quantiles). It is possible to visualize those two cases above, in green the area where risks are super-additive, while the yellow area is where risks are sub-additive.
Recall that with a Gaussian random vector, with correlation then the quantile is the quantile of a random variable centered, with variance
. Thus, on the graph below, we can visualize case that can be obtained with this Gaussian copula. Here the yellow area can be obtained with a Gaussian copulas, the upper and the lower bounds being respectively the comonotonic and the countermononic cases.
But the green area can also be obtained when we sum two Gaussian variables ! We just have to go outside the Gaussian world, and consider another copula.
Another point is that, in the previous post, was the lower Fréchet-Hoeffding bound on the set of copulas. But all the previous results remain valid if
is a lower bound on the set of copulas of interest. Especially
for all such that
. For instance, if we assume that the copula should have positive dependence, i.e.
, then
Which means we should have sharper bounds. Numerically, it is possible to compute those sharper bounds for quantiles. The lower bound becomes
while the upper bound is
Again, one can easily compute those quantities on a grid of the unit interval,
> Qinfind=Qsupind=rep(NA,n-1) > for(i in 1:(n-1)){ + J=1:(i) + Qinfind[i]=max(Finv(J/n)+Ginv((i-J)/n/(1-J/n))) + J=(i):(n-1) + Qsupind[i]=min(Finv(J/n)+Ginv(i/J)) + }
We get the graph below (the blue area is here to illustrate how sharper those bounds get with the assumption that we do have positive dependence, this area been attained only with copulas exhibiting non-positive dependence)
For high quantiles, the upper bound is rather close to the one we had before, since worst case are probably obtained when we do have positive correlation. But it will strongly impact the lower bound. For instance, it becomes now impossible to have a negative quantile, when the probability exceeds 75% if we do have positive dependence...
> Qinfind[u==.75] [1] 0
is Kendall'tau, and can be written
has copula 
and 
have the same distribution. Now, if we assume that those variables are (strictly) independent,
such that
is the survival copula associated with 
, in the sense that
from a Student-t distribution, 
can't be a copula, since margins are not even uniform on the unit interval. 
with distribution 
, define random variable 
. Then Kendall's cumulative function is
, compute 
as the proportion of observation in the lower quadrant, with upper corner 
, i.e.
, then Kendall's function is simply
, and the case of independence, the upper green curve, 
. It should also be mentioned that it is also common to plot not function 
, but function 
, defined as 
,
be a random pair with copula 
, and 
with copula 
. Then define
i.e.
. Here, computation can be tricky. Consider the following sample,
which leads to the following expressions
. Here,
denote the median of 
, i.e.
(e.g., as a start 
). From Fréchet (1951) or 
.
. In higher dimension, the following concept is necessary. A function 
is 2-increasing if its 
-volume 
is positive on any rectangle 
, where 
, and consider the 
, defined as
denote the parameter (of our parametric model, e.g. the tail index), and we would like to know whether 
(where in the context of finite versus infinite mean 
). I.e. either 
or to its complementary 
. Consider the maximum likelihood estimator 
, i.e.
and 
denote the constrained maximum likelihood estimators on 
and 
respectively,
and 
(on the left), or 
and 
(on the right)
