论文标题
凸变换β分布的订单和某些后果
Convex transform order of Beta distributions with some consequences
论文作者
论文摘要
凸变换顺序是对实际线上概率分布的偏度进行精确比较的一种方法。根据凸变换顺序，我们建立了一个beta分布小于另一个β分布时的简单而完整的表征。作为应用程序，我们得出单调性属性，以超过其分布的平均值或模式的β分布式随机变量的概率。此外，我们为单峰分布的模式中等不平等获得了一个简单的替代证明，该证明是通过凸变换顺序精确地倾斜的单峰分布的。这种新的证明还对具有独特抗模式的分布的抗模式产生了类似的不平等。 β分布的这种不平等作为特殊情况。最后，提到了在其平均值附近的二项式分布的分布函数值的某些后果。
The convex transform order is one way to make precise comparison between the skewness of probability distributions on the real line. We establish a simple and complete characterisation of when one Beta distribution is smaller than another according to the convex transform order. As an application, we derive monotonicity properties for the probability of Beta distributed random variables exceeding the mean or mode of their distribution. Moreover, we obtain a simple alternative proof of the mode-median-mean inequality for unimodal distributions that are skewed in a sense made precise by the convex transform order. This new proof also gives an analogous inequality for the anti-mode of distributions that have a unique anti-mode. Such inequalities for Beta distributions follow as special cases. Finally, some consequences for the values of distribution functions of Binomial distributions near to their means are mentioned.
