A differential equations approach to the modal location for a family of bivariate gamma distributions

Analytical and numerical results are given for determining the location of the mode of a class of bivariate gamma densities as a function of the parameters. The model location for a class of bivariate gammas as considered by Kibble (1941, Sankhya A 5 137–150) is shown to satisfy a nonlinear differential equation in ϱ, the correlation coefficient for fixed shape parameter. Qualitative and asymptotic properties of the modal location are also given. Whenever the shape parameters are unequal, analytical and numerical results are used to provide a conjecture for the modal location in the general case.