论文标题
二项式尾巴概率几乎紧密的通用界限
Nearly tight universal bounds for the binomial tail probabilities
论文作者
论文摘要
我们得出了适用于整个参数状态的二项式下尾概率的简单但几乎紧密的上限和下限。这些边界易于计算，并且在$ 89/44 $的恒定系数内紧密。此外，它们在较大偏差和中度偏差的机制中渐近地紧密。由于与Ramanujan的方程式有惊人的联系，我们还提供了有力的证据，表明下限紧密在1.26434美元的$ 1.26434 $之内。鉴于其简单性和吸引人的特性，它甚至可能被视为自然下限。我们的界限极大地超过了熟悉的Chernoff界限和反向Chernoff界限，并且可能在各个研究领域找到应用。
We derive simple but nearly tight upper and lower bounds for the binomial lower tail probability (with straightforward generalization to the upper tail probability) that apply to the whole parameter regime. These bounds are easy to compute and are tight within a constant factor of $89/44$. Moreover, they are asymptotically tight in the regimes of large deviation and moderate deviation. By virtue of a surprising connection with Ramanujan's equation, we also provide strong evidences suggesting that the lower bound is tight within a factor of $1.26434$. It may even be regarded as the natural lower bound, given its simplicity and appealing properties. Our bounds significantly outperform the familiar Chernoff bound and reverse Chernoff bounds known in the literature and may find applications in various research areas.
