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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:493

Article

Gaunt, Robert E.¹  Merkle, Milan² 

[1] Univ Manchester, Dept Math, Oxford Rd, Manchester M13 9PL, Lancs, England

[2] Univ Belgrade, Sch Elect Engn, Belgrade, Serbia

关键词: Generalized hyperbolic distribution;    Variance-gamma distribution;    McKay Type I distribution;    Mode;    Median;    Inequality;   

DOI  :  10.1016/j.jmaa.2020.124508

来源: Elsevier

【 摘 要 】

Except for certain parameter values, a closed form formula for the mode of generalized hyperbolic (GH) distribution is not available. In this paper, we exploit results from the literature on modified Bessel functions and their ratios to obtain simple but tight two-sided inequalities for the mode of the GH distribution 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 us to a conjectured tight lower bound for the median of the GH distribution. (c) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free