Abstract
Except for certain parameter values, a closed form formula for the mode of the
generalized hyperbolic (GH) distribution is not available. In this paper, we exploit
results from the literature on modied Bessel functions and their ratios to obtain
simple but tight two-sided inequalities for the mode of the GH distribution for general parameter values. As a special case, we deduce tight two-sided inequalities for the mode of the variance-gamma (VG) distribution, and through a similar approach we also obtain tight two-sided inequalities for the mode of the McKay Type I distribution. The analogous problem for the median is more challenging, but we conjecture some monotonicity results for the median of the VG and McKay Type I distributions, from we which we conjecture some tight two-sided inequalities for their medians. Numerical experiments support these conjectures and also lead us to a conjectured tight lower bound for the median of the GH distribution.
generalized hyperbolic (GH) distribution is not available. In this paper, we exploit
results from the literature on modied Bessel functions and their ratios to obtain
simple but tight two-sided inequalities for the mode of the GH distribution for general parameter values. As a special case, we deduce tight two-sided inequalities for the mode of the variance-gamma (VG) distribution, and through a similar approach we also obtain tight two-sided inequalities for the mode of the McKay Type I distribution. The analogous problem for the median is more challenging, but we conjecture some monotonicity results for the median of the VG and McKay Type I distributions, from we which we conjecture some tight two-sided inequalities for their medians. Numerical experiments support these conjectures and also lead us to a conjectured tight lower bound for the median of the GH distribution.
Original language | English |
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Journal | Journal of Mathematical Analysis and Applications |
Publication status | Accepted/In press - 11 Aug 2020 |
Keywords
- generalized hyperbolic distribution
- variance-gamma distribution
- McKay Type I distribution
- mode
- median
- inequality
- modified Bessel function