广义双曲分布及相关分布的众数和中位数的界
On bounds for the mode and median of the generalized hyperbolic and related distributions
摘要：
除某些参数值外，广义双曲（GH）分布的模式没有一个封闭形式的公式。在本文中，我们利用文献中的结果修改贝塞尔函数和他们的比率，以获得简单而紧密的双边不等式的模式GH分布的一般参数值。作为一种特殊情况，我们导出了方差-γ（VG）分布的模的紧双边不等式，并通过类似的方法得到了McKay I型分布的模的紧双边不等式.中位数的类似问题更具挑战性，但我们推测VG和McKay I型分布的中位数的单调性结果，从中我们推测其中位数的一些紧双边不等式。数值实验支持了这些假设，并使我们得到了GH分布中值的严格下界。
英文摘要：
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 modified 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.
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