Category Archives: Seminar

SOA Webinar on Predictive Modeling

I will give, with Qichun Xu, a joint webinar for the Reinsurance Council and the Futurism Council of the Society of Actuaries, on Perspectives of Predictive Modeling with Case Studies in a few days. The slides of my talk are now available (I do recommand to open the pdf version of the slides with Acrobat, since there are animated pictures in the slides that could not be visualized below for instance). The Society of Actuaries asked specifically for a powerpoint document, so I will use screenshots of the slides for the webinar. I do encourage to open and read the pdf file for a better quality… Sorry for the inconvenience. I will upload soon lines of codes to reproduce most of the graphs. All comments and remarks are welcome.

Beta kernel and transformed kernel

This Thursday I will give a talk at Laval University, on “Beta kernel and transformed kernel : applications to copula density estimation and quantile estimation“. This time, I will talk at the department of Mathematics and Statistics (13:30 at the pavillon Adrien-Pouliot). “Because copulas have bounded support (the unit square in dimension 2), standard kernel based estimators of densities are (multiplicatively) biased on borders and in corners of the support. Two techniques can be used to avoid that underestimation: Beta kernels and Transformed kernel. We will describe and discuss those two techniques in the first part of the talk. Then, we will see that it is possible to combine those two techniques to get nice estimator of several quantities (e.g. quantiles): transform the data to get on the unit interval – using a transformed kernel – then estimate the (transformed) quantile on [0,1] using a beta kernel, then get back on the initial support. As we will see on simulations, that technique can be better than standard quantile estimators, especially when data are heavy tailed.” Slides can be downloaded here.

  • kernel based density estimation

Kernel based estimation are a popular (and natural) technique to estimate densities.  It is simply and extension of the moving histogram:

so we count how many observations are a the neighborhood of the point where we want to estimate the density of the distribution. Then it is natural so consider a smoothing function, i.e. instead of a step function (either observations are close enough, or not), it is possible to give weights to observations, which will be a decreasing function of the distance,

With a smooth kernel, we have a smooth estimation of the density

http://freakonometrics.blog.free.fr/public/perso3/kernel-f-01.gif

Then it is possible to play on the bandwidth, either to get a more accurate estimation of the density, but not that smooth (small bias but large variance),

or a smoother one (large bias, but small variance),

In R, it is simply

> X=rnorm(100)
> (D=density(X))
 
Call:
	density.default(x = X)
 
Data: X (100 obs.);	Bandwidth 'bw' = 0.3548
 
       x                   y            
 Min.   :-3.910799   Min.   :0.0001265  
 1st Qu.:-1.959098   1st Qu.:0.0108900  
 Median :-0.007397   Median :0.0513358  
 Mean   :-0.007397   Mean   :0.1279645  
 3rd Qu.: 1.944303   3rd Qu.:0.2641952  
 Max.   : 3.896004   Max.   :0.3828215  
 
> plot(D$x,D$y)
  • Beta kernel

The idea of Beta kernel is to consider kernels having support [0,1]. In the univariate case,

http://freakonometrics.blog.free.fr/public/perso3/kernel-f-06.gif

where http://freakonometrics.blog.free.fr/public/perso3/kernel-f-07.gif is the density of a Beta distribution, i.e.

http://freakonometrics.blog.free.fr<br />
/public/perso3/beta-distribution.gif

For additional material, I have uploaded some R code to fit copula densities using beta kernels,

library(copula)
beta.kernel.copula.surface = function (u,v,bx,by,p) {
s = seq(1/p, len=(p-1), by=1/p)
mat = matrix(0,nrow = p-1, ncol = p-1)
for (i in 1:(p-1)) {
a = s[i]
for (j in 1:(p-1)) {
b = s[j]
mat[i,j] = sum(dbeta(a,u/bx,(1-u)/bx) *
dbeta(b,v/by,(1-v)/by)) / length(u)
} }
return(data.matrix(mat)) }

Then we can used it to see what we get on a simulated sample

library(copula)
COPULA = frankCopula(param=5, dim = 2)
X = rcopula(n=1000,COPULA)
p0 = 26
Z= beta.kernel.copula.surface(X[,1],X[,2],bx=.01,by=.01,p=p0)
u = seq(1/p0, len=(p0-1), by=1/p0)
persp(u,u,Z,theta=30,col="green",shade=TRUE,
box=FALSE,zlim=c(0,6))

http://freakonometrics.free.fr/copula-kernel-beta.gif
(yes, the surface is changing… to illustrate the impact of the bandwidth on the estimation).

  • transformed kernel estimation

I the talk, I will also mention the transformed Kernel estimate, as introduced in the book on L1 density estimation by Luc Devroye and Laszlo Györfi (the book can be downloaded here). I probably spend a few minutes on the original chapter, in order to provide another application of that techniques (not only to estimate copula densities, but here to estimate quantiles of heavy tailed distribution). In the univariate case, the R code is the following (here I consider two transformation, the quantile function of the Gaussian distribution, and the quantile function of the Student distribution with 3 degrees of freedom),

set.seed(1)
sample=rbeta(100,4,3)
 
transfN = function(x){
Y=qnorm(sample)
f=density(Y,from=-4,to=4,n=2001)
ny=sum(f$x<=qnorm(x)); 
  g=f$y[ny]/dnorm(qnorm(x))
return(g)
}
 
df0=3
 
transfT = function(x){
Y=qt(sample,df=df0)
f=density(Y,from=-4,to=4,n=2001)
ny=sum(f$x<=qt(x,3)); 
  g=f$y[ny]/dt(qt(x,df=df0),df=df0)
return(g)
}
 
tN=Vectorize(transfN)
tT=Vectorize(transfT)
 
u=seq(.01,.99,by=.01)
vN=tN(u)
vT=tT(u)
plot(u,vN,type="l",lwd=3,col="blue")
lines(u,vT,lwd=3,col="green")
lines(u,dbeta(u,4,3),col="red",lty=2)

The density estimation is the following,

(the red dotted line is the true density, since we work on a simulated sample). Now, let us get back on the initial chapter,

In the book, this is introduced as follows,

The original idea we add it to use this kernel based estimator for copulas, i.e. since we can estimate densities in high dimension with unbounded support, using

http://freakonometrics.blog.free.fr/public/perso3/kernel-f-02.gif

the idea is to transform marginal observations,

http://freakonometrics.blog.free.fr/public/perso3/kernel-f-10.gif

and to use the fact that the associated copula density can be written

http://freakonometrics.blog.free.fr/public/perso3/kernel-f-12.gif

to derive an intuitive estimator for the copula density

http://freakonometrics.blog.free.fr/public/perso3/kernel-f-13.gif

An important issue is how do we choose the transformation

And Luc Devroye and Laszlo Györfi mention that this can be used to deal with extremes.

well, extremes are introduced through bumps (which is not the way I would have been dealing with extremes)

and note that several results can be derived on those bumps,

e.g.

Then, there is an interesting discussion about estimating the optimal transformation

and I will prove that this can be an extremely interesting idea, for instance to estimate quantiles of heavy tailed distribution, if we use also the beta kernel estimator on the unit interval. This idea was developed in a paper with Abder Oulidi, online here.

Remark: actually, in the book, an additional reference is mentioned,

but I have never been able to find a copy… if anyone has one, I’d be glad to read it…

Séminaire Probabilité et Statistique, UBO, Brest

Talk at the statistical seminar at the Université de Bretagne Occidentale, in Brest, Wednesday May 6th Tuesday May 5th, 14h (in  10 days), on “multivariate extremes. Slides can be found here.

The talk will give a detailed introduction on multivariate extremes and related concepts. Then the case of Archimedean copula will be fully described (following the paper with Johan Segers).

[04/05/2009]: some applications in risk management will be shown at the end of talk, as well as some news things on spatial correlation.

and in order to illustrate tail convergence of Archimedean copulas, I have uploaded two animations, with tail independence below,

with tail dependence (or asymptotic dependence),

Workshop in Sao Paulo

Talk at the workshop in Sao Paulo, Thursday, on “estimation of quantile related risk measures“. The workshop also invited Claudia Kluppelberg, Richard Davis, and  Ermanno Pitacco to talk. Slides can be found here. And maybe to explain a bit more where this idea of beta-kernels and transformed kernel comes from, I should mention those slides (see here for a more detailed version of the slides, and there for the full version of the paper with Jean David Fermanian and Olivier Scaillet).

Statistical seminar at Belo Horizonte

Talk at the statistical seminar at the university of Belo Horizonte, Wednesday, onmultivariate extremes. Slides can be downloaded here.

The talk will give a detailed introduction on multivariate extremes and related concepts. Then the case of Archimedean copula will be fully described (following the paper with Johan Segers).

Many thanks to Renato Martins Assunção (here) for inviting me for a couple of days in Belo Horizonte ! Thanks also for your interest in my blog… and since I understood that some people who do not speak French might be interested in my blog, I started to write my blog in English (or at least a langage that should not be too far away from English). There is a nice discussion about langage on this blog (here, unfortunately in French…)

Exposé à Toulouse sur l’estimation (nonparamétrique) de quantiles

Exposé à Toulouse 1, sur l’estimation nonparamétrique de quantiles.

In this talk we propose several nonparametric estimators of quantiles based on Beta kernel and applied to transformed data by the generalized Champernowne distribution initially fitted to the data. A Monte-Carlo based study will show that those estimators improve the efficiency, not only for light tailed distributions, but mainly for heavy tailed, when the probability level is close to 1. Another application will be seen, on portfolio optimization in the mean-VaR context.