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(This article was first published on <strong><a href="https://xianblog.wordpress.com/2016/07/04/correlation-matrices-on-copulas/"> R – Xi'an's Og</a></strong>, and kindly contributed to <a rel="nofollow" href="http://www.r-bloggers.com/">R-bloggers)</a> </div>
<p><strong><a target="_blank" href="http://i2.wp.com/xianblog.files.wordpress.com/2016/06/letac.jpg?ssl=1"><img alt="" class="wp-image-30971" src="http://i2.wp.com/xianblog.files.wordpress.com/2016/06/letac.jpg?resize=450%2C253&ssl=1" data-recalc-dims="1"></a><a target="_blank" href="http://proba.jussieu.fr/~jmyor"><img height="262" width="185" src="https://i1.wp.com/proba.jussieu.fr/~jmyor/docs/Affiche_Yor_mini.png"></a>F</strong>ollowing my <a target="_blank" href="https://xianblog.wordpress.com/2016/06/27/another-wrong-entry/">post of yesterday</a> about the missing condition in <a target="_blank" href="https://xianblog.wordpress.com/2015/07/18/the-worst-possible-proof-xed/">Lynch’s</a> R code, Gérard Letac sent me a paper he recently wrote with Luc Devroye on correlation matrices and copulas. Paper written for the <a target="_blank" href="http://link.springer.com/chapter/10.1007/978-3-319-18585-9_25">memorial volume</a> in honour of <a target="_blank" href="https://xianblog.wordpress.com/2015/05/14/marc-yor/">Marc Yor</a>. 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</p>
<p><img title="r^0_{ij}=\frac{6}{\pi}\arcsin\{r_{ij}/2\}" alt="r^0_{ij}=\frac{6}{\pi}\arcsin\{r_{ij}/2\}" class="latex" src="https://s0.wp.com/latex.php?latex=r%5E0_%7Bij%7D%3D%5Cfrac%7B6%7D%7B%5Cpi%7D%5Carcsin%5C%7Br_%7Bij%7D%2F2%5C%7D&bg=000000&%23038;fg=B0B0B0&%23038;s=0"></p>
<p>A side consequence of this result is that there exist correlation matrices of copulas that cannot be associated with Gaussian copulas. Like</p>
<p><img title="R=\left[\begin{matrix} 1 &amp;-1/2 &amp;-1/2\\-1/2 &amp;1 &amp;-1/2\\-1/2 &amp;-1/2 &amp;1 \end{matrix}\right]" alt="R=\left[\begin{matrix} 1 &amp;-1/2 &amp;-1/2\\-1/2 &amp;1 &amp;-1/2\\-1/2 &amp;-1/2 &amp;1 \end{matrix}\right]" class="latex" src="https://s0.wp.com/latex.php?latex=R%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D+1+%26-1%2F2+%26-1%2F2%5C%5C-1%2F2+%261+%26-1%2F2%5C%5C-1%2F2+%26-1%2F2+%261+%5Cend%7Bmatrix%7D%5Cright%5D&bg=000000&%23038;fg=B0B0B0&%23038;s=0"></p>
<p>Filed under: <a target="_blank" href="https://xianblog.wordpress.com/category/statistics/r-statistics/">R</a>, <a target="_blank" href="https://xianblog.wordpress.com/category/statistics/">Statistics</a>, <a target="_blank" href="https://xianblog.wordpress.com/category/university-life/">University life</a> Tagged: <a target="_blank" href="https://xianblog.wordpress.com/tag/copulas/">copulas</a>, <a target="_blank" href="https://xianblog.wordpress.com/tag/correlation/">correlation</a>, <a target="_blank" href="https://xianblog.wordpress.com/tag/gerard-letac/">Gérard Letac</a>, <a target="_blank" href="https://xianblog.wordpress.com/tag/luc-devroye/">Luc Devroye</a>, <a target="_blank" href="https://xianblog.wordpress.com/tag/marc-yor/">Marc Yor</a> <a target="_blank" rel="nofollow" href="http://feeds.wordpress.com/1.0/gocomments/xianblog.wordpress.com/30969/"><img alt="" src="http://feeds.wordpress.com/1.0/comments/xianblog.wordpress.com/30969/" border="0"></a> <img alt="" src="http://i1.wp.com/pixel.wp.com/b.gif?resize=1%2C1&ssl=1" border="0" data-recalc-dims="1"></p>
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