# In statistics, having too much information might not be a good thing

A common idea in statistics is that if we don’t know something, and we use anestimator of that something (instead of the true value) then there will be some additional uncertainty. For instance, consider a random sample, i.i.d., from a Gaussian distribution. Then, a confidence interval for the mean is

where  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.

and the cost we have to pay is that the new confidence interval is

where now  is the quantile of the Student distribution, of probability level , with  degrees of freedom.
We call it a cost since the new confidence interval is now larger (the Student distribution has higher upper-quantiles than the Gaussian distribution).
So usually, if we substitute an estimation to the true value, there is a price to pay.
A few years ago, with Jean David Fermanian and Olivier Scaillet, we were writing a survey on copula density estimation (using kernels,  here). At the end, we wanted to add a small paragraph on the fact that we assumed that we wanted to fit a copula on a sample  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 ). But since margins are usually unknown, there should be a price for not observing them.
To be more formal, in a perfect wold, we would consider

but in the real world, we have to consider

where it is standard to consider ranks, i.e.  are empirical cumulative distribution functions.
My point is that when I ran simulations for the survey (the idea was more to give illustrations of several techniques of estimation, rather than proofs of technical theorems) we observed that the price to pay… was negative ! I.e. the variance of the estimator of the density (wherever on the unit square) was smaller on the pseudo sample  than on perfect sample .
By that time, we could not understand why we got that counter-intuitive result: even if we do know the true distribution, it is better not to use it, and to use instead a nonparametric estimator. Our interpretation was based on the discrepancy concept and was related to the latin hypercube construction:

With ranks, the data are more regular, and marginal distributions are exactlyuniform on the unit interval. So there is less variance.
This was our heuristic interpretation.
A couple of weeks ago, Christian Genest and Johan Segers proved that intuition in an article published in JMVA,

Well, we observed something for finite , but Christian and Johan obtained an analytical result. Hence, if we denote

the empirical copula in the perfect world (with known margins) and

the one constructed from the pseudo sample, they obtained that, everywhere

with nice graphs of ,

So I was very happy last week when Christian show me their results, to learn that our intuition was correct. Nevertheless, it is still a very counter-intuitive result…. If anyone has seen similar things, I’d be glad to hear about it !

# 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…)

# Tails of Archimedean copulas

Publication de Tails of Archimedean Copulas, coécrit avec Johan Segers, dans le Journal of Multivariate Analysis, enfin… en ligne sur http://sciencedirect.com/science…

A complete and user-friendly directory of tails of Archimedean copulas is presented which can be used in the selection and construction of appropriate models with desired properties. The results are synthesized in the form of a decision tree: Given the values of some readily computable characteristics of the Archimedean generator, the upper and lower tails of the copula are classified into one of three classes each, one corresponding to asymptotic dependence and the other two to asymptotic independence. For a long list of single-parameter families, the relevant tail quantities are computed so that the corresponding classes in the decision tree can easily be determined. In addition, new models with tailor-made upper and lower tails can be constructed via a number of transformation methods. The frequently occurring category of asymptotic independence turns out to conceal a surprisingly rich variety of tail dependence structures.

# Convergence of Archimedean Copulas

The paper on Convergence of Archimedean Copulas, with Johan Segers, just appeared, in Statistics and Probability Letters.

Convergence of a sequence of bivariate Archimedean copulas to another Archimedean copula or to the comonotone copula is shown to be equivalent with convergence of the corresponding sequence of Kendall distribution functions. No extra differentiability conditions on the generators are needed.