On bounds for the mode and median of the generalized hyperbolic and related distributions

Except for certain parameter values, a closed form formula the mode of generalized hyperbolic (GH) distribution is not available. In this paper, we exploit results from literature on modified Bessel functions and their ratios to obtain simple but tight two-sided inequalities GH general values. As special case, deduce variance-gamma (VG) distribution, through similar approach also McKay Type I distribution. The analogous problem median more challenging, conjecture some monotonicity VG distributions, which medians. Numerical experiments support these conjectures lead us conjectured lower bound