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.