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 alower 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
where 
For instance, if
then, we obtain Gumbel copula. This is what we’ve seen in the section where we introduced this family. Now, let us talk about (nonparametric) inference, and more precisely the estimation of the dependence function. The starting point of the most standard estimator is to observe that if 
has distribution function
And conversely, Pickands dependence function can be written
This is the estimator proposed in 
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
If we’re too lazy to do the maths, at least, it is possible to compute those functions numerically. For instance, for Clayton copula,
, 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 
,
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)
. Let 
denote the Gamma distribution with density (on 
)
denote independent 
random variables, with 
. Then 
where
has a distribution in the simplex of 
.
can be visualized below, e.g. 
, with some kind of symmetry,
and 
, below
, and if 
, then 
‘s have Beta distributions,
(either the company defaults, or not), so that
is said to be exchangeable if for all 
for any permutation 
of 
. A standard example is the case where
, with
and
Since 
, a necessary condition is that
i.e.
.
. 
on 
such that
where 
. Note that 
A nice proof of that result can be found in 
of 
,
that can be inverted in
The idea is then to extend the size of the vector 
, i.e. for all 
, define
so that, if we condition on 
,
but since given the sum of components of 
elements are equally likely, we can write
The first idea is to work on the blue term, and to invocate a theorem of approximation of the hypergeometric distribution 
to a binomial distribution 
, when 
becomes large. Then
Let 
and let 
denote the cumulative distribution function of 
.
The idea is then to write the sum as an integral, with respect to that distribution,
The theorem is then obtained since 
, i.e.
In the case of non-binary sequences, there is an extension of the previous result,
. 
on 
such that
where 
and
For instance, in the Gaussian case mentioned earlier, if
then
where
i.e. conditionally on 
, the 
. The proof can be found in 
where 
are i.i.d. random variables. To go further on exchangeability and related topics, see 
Since
if we assume that – given the latent factor – 
(either the company defaults, or not),
i.e.
Thus, we can derive the (unconditional) distribution of the sum
i.e.
. In the upper corner is plotted the density of the Beta distribution.
of 
random variables is exchangeable if and only if 
‘s are conditionnally independent, conditionnally on some random variable 
.
, a Generalized Pareto Distribution will fit nicely, then we can use it to derive an estimator of the quantile function (for percentages such that the quantile is larger than the threshold)
, i.e. we keep the 
largest observations to fit a GPD, then this estimator can be written
, where 
is a slowly varying function. Then, for all 
,
exceed the threshold out of a sample of size 
. Then
