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Bounds for the median of the generalized hyperbolic and related distributions

  • Robert E. Gaunt
  • Frédéric Ouimet
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Abstract

We prove monotonicity properties for medians of gamma sums and differences of the form Zα = αX1 + (2 − α)X2, where X1 and X2 are independent gamma random variables with common shape parameter. By combining these monotonicity properties with known bounds for the median of the gamma distribution, we establish sharp bounds for the median of the variance-gamma and McKay Type I distributions. Also, by exploiting the normal variance-mean mixture representation of the generalized hyperbolic distribution together with bounds for a ratio of modified Bessel functions of the second kind, we obtain sharp bounds for the median of the generalized hyperbolic distribution. We thus resolve all five conjectures of Gaunt and Merkle (2021). As a by-product of our analysis, we show that the variance-gamma distributions with positive asymmetry parameter and the McKay Type I distributions satisfy the ``mode-median-mean'' inequality for all admissible parameter values, and that the same is true of the generalized hyperbolic distribution with positive asymmetry parameter.

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Cite

BibTeXRIS
BibTeX
@misc{gaunt2026bounds,
  title = {{Bounds for the median of the generalized hyperbolic and related distributions}},
  author = {Robert E. Gaunt and Frédéric Ouimet},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20212},
  eprint = {2609.20212},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20212},
}
RIS
TY  - UNPB
TI  - Bounds for the median of the generalized hyperbolic and related distributions
AU  - Robert E. Gaunt
AU  - Frédéric Ouimet
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.20212
UR  - https://arxiv.org/abs/2609.20212
AB  - We prove monotonicity properties for medians of gamma sums and differences of the form Z_α = αX_1 + (2 − α)X_2, where X_1 and X_2 are independent gamma random variables with common shape parameter. By combining these monotonicity properties with known bounds for the median of the gamma distribution, we establish sharp bounds for the median of the variance-gamma and McKay Type I distributions. Also, by exploiting the normal variance-mean mixture representation of the generalized hyperbolic distribution together with bounds for a ratio of modified Bessel functions of the second kind, we obtain sharp bounds for the median of the generalized hyperbolic distribution. We thus resolve all five conjectures of Gaunt and Merkle (2021). As a by-product of our analysis, we show that the variance-gamma distributions with positive asymmetry parameter and the McKay Type I distributions satisfy the ``mode-median-mean'' inequality for all admissible parameter values, and that the same is true of the generalized hyperbolic distribution with positive asymmetry parameter.
ER  -