论文标题
阈值的隐藏方程
Hidden Equations of Threshold Risk
论文作者
论文摘要
我们考虑阈值风险的敏感性问题，定义为一个随机变量函数的概率低于指定的阈值水平$δ>0。$ $我们证明，对于该随机变量的多项式和合理函数，在最有限的风险关键点处存在。后者是阈值参数的特殊值，因为$δ$接近这些阈值值，风险变化率无限。我们将风险关键点的候选人表征为相关$δ-$扰动多项式的分解或其领先系数或两者兼而有之的零。因此，需要求解的方程本身就是$δ$中的多项式方程，这些方程利用了基础多项式或合理函数的代数特性。我们将这些重要方程式称为“阈值风险的隐藏方程”。
We consider the problem of sensitivity of threshold risk, defined as the probability of a function of a random variable falling below a specified threshold level $δ>0.$ We demonstrate that for polynomial and rational functions of that random variable there exist at most finitely many risk critical points. The latter are those special values of the threshold parameter for which rate of change of risk is unbounded as $δ$ approaches these threshold values. We characterize candidates for risk critical points as zeroes of either the resolvent of a relevant $δ-$perturbed polynomial, or of its leading coefficient, or both. Thus the equations that need to be solved are themselves polynomial equations in $δ$ that exploit the algebraic properties of the underlying polynomial or rational functions. We name these important equations as "hidden equations of threshold risk".
