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Given the variables $X$ and $Y$, which are correlated, $X\ge0$, $Y\ge0$ and each follow a gamma distribution with different shape parameters, i.e.,$X\sim Gamma(a_1,\alpha)$ and $Y\sim Gamma(a_2,\alpha)$.

I understood that the Joint PDF $f_{X,Y}(x,y)$ can be obtained by making use of the Farlie Gumbel Morgenstern (FGM) Bivariate Gamma Copula which is applicable for the case of different shape parameters. The joint PDF is given by

$$h_{X,Y}(x,y) = f_X(x)f_Y(y) \left[1+\lambda\left(2F_X(x)-1\right)\left(2F_Y(y)-1\right)\right],\quad |\lambda|\le 1 $$

where $f(.)$ denotes the marginal PDF and $F(.)$ denotes the marginal CDF. My major questions are:

  • How to determine the value of $\lambda$?
  • Is it true that the FGM copula is used only for weak correlation?
  • Is there another method of calculating Pearson's correlation coefficient rather than the equation given by $$ \rho = \lambda K(a_1)K(a_2)$$ where $K(a)=\displaystyle\frac{1}{2^{2a-1} \beta(a,a) \sqrt{a}}$ and $\beta(.,.)$ is the beta function.
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    $\begingroup$ Two common ways to estimate parameters like $\lambda$ in that copula would be ML or by matching the sample and the population nonparametric correlation implied by $\lambda$ (Spearman rho or Kendall tau) and backing the corresponding $\lambda$ out as an estimate. These are far from the only possibilities. $\endgroup$
    – Glen_b
    Nov 20, 2012 at 23:31
  • $\begingroup$ Thanks Glen_b, I understand about maximum likelihood estimators, can you please explain more about the second method you suggested? $\endgroup$
    – Remy
    Nov 21, 2012 at 17:30
  • $\begingroup$ Okay. It's a bit long for a comment, so I'll move it to an answer. $\endgroup$
    – Glen_b
    Nov 21, 2012 at 22:19
  • $\begingroup$ NB: See eqn (5) here $\endgroup$
    – Glen_b
    Sep 30, 2015 at 3:22

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To expand on my comment about matching the sample and the population nonparametric correlation implied by $\lambda$ (Spearman's $\rho$ or Kendall's $\tau$) and backing the corresponding $\lambda$ out as an estimate.

I guess Nelsen's book would be a starting point but you can get some stuff on it elsewhere.

For example, check out:

Christian Genest and Anne-Catherine Favre, "Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask", Journal of Hydrologic Engineering, July/August 2007.

(Which is available in pdf form online - e.g. via citeseer - google should find it. In particular, check the section on Estimation)

If I have it right (and I may not) for the FGM copula they give $c(u,v) = 1 + \theta (1 − 2u)(1 − 2v)$ and from that compute the population Kendall's $\tau$ as $2\theta/9$ (and $\rho$ as $\theta/3$). So you take a sample, compute the sample Kendall's $\tau$, $\tau_n$ and equate that to $2\theta/9$, from which you'd estimate $\theta$ as $4.5 \tau_n$.

Similarly with Spearman's $\rho$, but there $\theta$ would be estimated as $3 \rho_n$.

A bunch of other papers and books describe this approach. Here's another example I just found (see Sec 3.4 there - but there they call the parameter $\alpha$).

Hope that helps.

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