Gamma分布中位数的闭式紧界与近似
Closed-form Tight Bounds and Approximations for the Median of a Gamma Distribution
AI总结：
本文提出在形状参数 $k > 0$ 全范围内寻找 Gamma 分布中位数 $2^{-1/k} (A + Bk)$ 形式的最紧闭式上下界，并通过插值和近似获得更紧的界及高精度近似公式。
摘要：
我们展示了如何在形状参数 $k > 0$ 的整个范围内，寻找 Gamma 分布中位数的上界和下界，这些界是 $2^{-1/k} (A + Bk)$ 形式的最紧可能界，其中 $A$ 和 $B$ 为闭式参数。这种形式下在高 $k$ 处表现最优的下界保持在第 48 至第 50 百分位之间，而唯一最优的上界保持在第 50 至第 55 百分位之间。我们展示了如何通过在这些界之间进行插值来形成更紧的界，从而产生更紧密地界定中位数的闭式表达式。此外，还找到了界之间的良好闭式近似，其中一种在 $k = 1$ 处是精确的，并保持在第 49.97 至第 50.03 百分位之间。
英文摘要：
We show how to find upper and lower bounds to the median of a gamma distribution, over the entire range of shape parameter $k > 0$, that are the tightest possible bounds of the form $2^{-1/k} (A + Bk)$, with closed-form parameters $A$ and $B$. The lower bound of this form that is best at high $k$ stays between 48 and 50 percentile, while the uniquely best upper bound stays between 50 and 55 percentile. We show how to form even tighter bounds by interpolating between these bounds, yielding closed-form expressions that more tightly bound the median. Good closed-form approximations between the bounds are also found, including one that is exact at $k = 1$ and stays between 49.97 and 50.03 percentile.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
