3

Let \begin{equation} c(u_1,u_2|k) = k\,\big((1-u_1)\,(1-u_2)\big)^{k-1}\, _2F_1\!\left(1-k,1-k;1; \frac{u_1\,u_2}{(1-u_1)\,(1-u_2)}\right) , \end{equation} where $k \in \{1, 2, \ldots\}$ and where $_2F_1$ is the hypergeometric function. The correlation between $u_1$ and $u_2$ is \begin{equation} \frac{k-1}{k+1} . \end{equation} Does this copula have a name? Where can I read about it?

mef
  • 3,226
  • Since apparently $k$ is a positive integer, it might help to recognize that $$,_2F_1(1-k,1-k,1,x) = 1 + \binom{k-1}{1}^2x + \binom{k-1}{2}^2 x^2 + \cdots + \binom{k-1}{k-2}^2x^{k-2}+x^{k-1}.$$ – whuber Nov 23 '18 at 19:04

1 Answers1

1

It turns out this is an example of and order-statistic-based copula. You can read about them here:

An order-statistics-based method for constructing multivariate distributions with fixed marginals

Here is the abstract, which provides a good summary:

A new system of multivariate distributions with fixed marginal distributions is introduced via the consideration of random variates that are randomly chosen pairs of order statistics of the marginal distributions. The distributions allow arbitrary positive or negative Pearson correlations between pairs of random variates and generalise the Farlie–Gumbel–Morgenstern distribution. It is shown that the copulas of these distributions are special cases of the Bernstein copula. Generation of random numbers from the distributions is described, and formulas for the Kendall and grade (Spearman) correlations are given. Procedures for data fitting are described and illustrated with examples.

By replacing $u_1$ with $1-u_1$ one obtains the a copula with negative correlation. With $k = 2$, the copula in the question is the FGM copula with maximal correlation.

mef
  • 3,226