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

Robert Gaunt, Milan Merkle

Research output: Contribution to journalArticlepeer-review

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.
Original languageEnglish
JournalJournal of Mathematical Analysis and Applications
Publication statusAccepted/In press - 11 Aug 2020

Keywords

  • generalized hyperbolic distribution
  • variance-gamma distribution
  • McKay Type I distribution
  • mode
  • median
  • inequality
  • modified Bessel function

Fingerprint

Dive into the research topics of 'On bounds for the mode and median of the generalized hyperbolic and related distributions'. Together they form a unique fingerprint.

Cite this