论文标题
gamma分布中位数的封闭形式的紧密界限和近似值
Closed-form Tight Bounds and Approximations for the Median of a Gamma Distribution
论文作者
论文摘要
我们展示了如何在整个形状参数$ k> 0 $的范围内找到上层和下限，这是表格$ 2^{ - 1/k}（a + bk）$的最紧可能的界限，带有封闭形式参数$ a $ a $ a $ a $ a $和$ b $。该表格的下限最佳$ k $在48至50％之间保持不变，而独特的最佳上限保持在50％至55％之间。我们通过在这些边界之间进行插值来展示如何形成更紧密的边界，从而产生更紧密地绑定中位数的封闭形式表达式。还发现了边界之间的良好封闭形式近似，其中包括$ k = 1 $，并且保持在49.97至50.03％之间。
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.
