For the last course MAT8886 of this (long) winter session, on copulas (and extremes), we will discuss risk aggregation. The course will be mainly on the problem of bounding the distribution (or some risk measure, say the Value-at-Risk) for two random variables with given marginal distribution. For instance, we have two Gaussian risks. What could be be worst-case scenario for the 99% quantile of the sum ? Note that I mention implications in terms of risk management, but of course, those questions are extremely important in terms of statistical inference, see e.g. Fan & Park (2006).
This problem, is sometimes related to some question asked by Kolmogorov almost one hundred years ago, as mentioned in Makarov (1981). One year after, Rüschendorf (1982) also suggested a proof of bounds calculation. Here, we focus in dimension 2. As usual, it is the simple case. But as mentioned recently, in Kreinovich & Ferson (2005), in dimension 3 (or higher), “computing the best-possible bounds for arbitrary n is an NP-hard (computationally intractable) problem“. So let us focus on the case where we sum (only) two random variable (for those interested in higher dimension, Puccetti & Rüschendorf (2012) provided interesting results for a dual version of those optimal bounds).
Let denote the set of univariate continuous distribution function, left-continuous, on
. And
the set of distributions on
. Thus,
if
and
. Consider now two distributions
. In a very general setting, it is possible to consider operators on
. Thus, let
denote an operator, increasing in each component, thus that
. And consider some function
assumed to be also increasing in each component (and continuous). For such functions
and
, define the following (general) operator,
as
One interesting case can be obtained when is a copula,
. In that case,
and further, it is possible to write
It is also possible to consider other (general) operators, e.g. based on the sum
or on the minimum,
where is the survival copula associated with
, i.e.
. Note that those operators can be used to define distribution functions, i.e.
and similarly
All that seems too theoretical ? An application can be the case of the sum, i.e. , in that case
is the distribution of sum of two random variables with marginal distributions
and
, and copula
. Thus,
is simply the convolution of two distributions,
The important result (that can be found in Chapter 7, in Schweizer and Sklar (1983)) is that given an operator , then, for any copula
, one can find a lower bound for
as well as an upper bound
Those inequalities come from the fact that for all copula ,
, where
is a copula. Since this function is not copula in higher dimension, one can easily imagine that get those bounds in higher dimension will be much more complicated…
In the case of the sum of two random variables, with marginal distributions and
, bounds for the distribution of the sum
, where
and
, can be written
for the lower bound, and
for the upper bound. And those bounds are sharp, in the sense that, for all , there is a copula
such that
and there is (another) copula such that
Thus, using those results, it is possible to bound cumulative distribution function. But actually, all that can be done also on quantiles (see Frank, Nelsen & Schweizer (1987)). For all let
denotes its generalized inverse, left continuous, and let
denote the set of those quantile functions. Define then the dual versions of our operators,
and
Those definitions are really dual versions of the previous ones, in the sense that and
.
Note that if we focus on sums of bivariate distributions, the lower bound for the quantile of the sum is
while the upper bound is
A great thing is that it should not be too difficult to compute numerically those quantities. Perhaps a little bit more for cumulative distribution functions, since they are not defined on a bounded support. But still, if the goal is to plot those bounds on , for instance. The code is the following, for the sum of two lognormal distributions
.
> F=function(x) plnorm(x,0,1) > G=function(x) plnorm(x,0,1) > n=100 > X=seq(0,10,by=.05) > Hinf=Hsup=rep(NA,length(X)) > for(i in 1:length(X)){ + x=X[i] + U=seq(0,x,by=1/n); V=x-U + Hinf[i]=max(pmax(F(U)+G(V)-1,0)) + Hsup[i]=min(pmin(F(U)+G(V),1))}
If we plot those bounds, we obtain
> plot(X,Hinf,ylim=c(0,1),type="s",col="red") > lines(X,Hsup,type="s",col="red")

But somehow, it is even more simple to work with quantiles since they are defined on a finite support. Quantiles are here
> Finv=function(u) qlnorm(u,0,1) > Ginv=function(u) qlnorm(u,0,1)
The idea will be to consider a discretized version of the unit interval as discussed in Williamson (1989), in a much more general setting. Again the idea is to compute, for instance
The idea is to consider and
, and the bound for the quantile function at point
is then
The code to compute those bounds, for a given is here
> 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)) + }
Here we have (several s were considered, so that we can visualize the convergence of that numerical algorithm),

Here, we have a simple code to visualize bounds for quantiles for the sum of two risks. But it is possible to go further…





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
,
. 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 the quantile of probability level
of the standard normal distribution
. But usually, standard deviation
(the something is was talking about earlier) is usually unknown. So we substitute an estimation of the standard deviation, e.g.


is the quantile of the Student distribution, of probability level
degrees of freedom.
i.i.d. with distribution
, a copula, but in practice, we start from a sample
with joint distribution
(assumed to have continuous margins, and – unique – copula 

are empirical cumulative distribution functions.
than on perfect sample
.

, but Christian and Johan obtained an analytical result. Hence, if we denote


,





et
ont la même loi et que l’on suppose les variables indépendantes, alors

tel que




est la copule de survie associée à
, au sens où














