## coupling, donkeys, coins & fish meet in Paris

Posted in Statistics with tags , , , , , , , , , , , , , , , , , , , , , , on March 22, 2021 by xi'an

## correlation matrices on copulas

Posted in R, Statistics, University life with tags , , , , on July 4, 2016 by xi'an

Following my post of yesterday about the missing condition in Lynch’s R code, Gérard Letac sent me a paper he recently wrote with Luc Devroye on correlation matrices and copulas. Paper written for the memorial volume in honour of Marc Yor. It considers the neat problem of the existence of a copula (on [0,1]x…x[0,1]) associated with a given correlation matrix R. And establishes this existence up to dimension n=9. The proof is based on the consideration of the extreme points of the set of correlation matrices. The authors conjecture the existence of (10,10) correlation matrices that cannot be a correlation matrix for a copula. The paper also contains a result that answers my (idle) puzzling of many years, namely on how to set the correlation matrix of a Gaussian copula to achieve a given correlation matrix R for the copula. More precisely, the paper links the [correlation] matrix R of X~N(0,R) with the [correlation] matrix R⁰ of Φ(X) by

$r^0_{ij}=\frac{6}{\pi}\arcsin\{r_{ij}/2\}$

A side consequence of this result is that there exist correlation matrices of copulas that cannot be associated with Gaussian copulas. Like

$R=\left[\begin{matrix} 1 &-1/2 &-1/2\\-1/2 &1 &-1/2\\-1/2 &-1/2 &1 \end{matrix}\right]$

## Laplace great⁶-grand child!

Posted in Kids, pictures, Statistics, University life with tags , , , , , , , , , on August 3, 2015 by xi'an

Looking at the Family Tree application (I discovered via Peter Coles’ blog), I just found out that I was Laplace’s [academic] great-great-great-great-great-great-great-grand-child! Through Poisson and Chasles. Going even further, as Simeon Poisson was also advised by Lagrange, my academic lineage reaches Euler and the Bernoullis. Pushing always further, I even found William of Ockham along one of the “direct” branches! Amazing ancestry, to which my own deeds pay little homage if any… (However, I somewhat doubt the strength of the links for the older names, since pursuing them ends up at John the Baptist!)

I wonder how many other academic descendants of Laplace are alive today. Too bad Family Tree does not seem to offer this option! Given the longevity of both Laplace and Poisson, they presumably taught many students, which means a lot of my colleagues and even of my Bayesian colleagues should share the same illustrious ancestry. For instance, I share part of this ancestry with Gérard Letac. And both Jean-Michel Marin and Arnaud Guillin. Actually, checking with the Mathematics Genealogy Project, I see that Laplace had… one student!, but still a grand total of [at least] 85,738 descendants… Incidentally, looking at the direct line, most of those had very few [recorded] descendants.