The maximum distribution of Kibble's bivariate gamma random vector

The joint probability density function of a random vector (Y1, Y2) following Kibble’s bivariate gamma distribution (BGD) with parameters α, μ1, μ2 and ρ, is given by fY1,Y2(y1, y2) = (μ1μ2) α (1− ρ)Γ(α) ( y1y2 ρμ1μ2 )α−1 2 exp ( − μ1y1 + μ2y2 1− ρ ) Iα−1 ( 2 √ ρμ1μ2y1y2 1− ρ ) , for y1 > 0, y2 > 0, where Γ(α) is the gamma function defined as Γ(α) = ∫∞ 0 t α−1e−tdt, and Iα(·) is the modified Bessel function of the first kind of order α defined as