Hidden Equations of Threshold Risk

TL;DR

This paper investigates the sensitivity of threshold risk for polynomial and rational functions, revealing that critical points are finitely many and characterized by solving specific algebraic equations called "hidden equations of threshold risk."

Contribution

It introduces the concept of hidden equations to identify risk critical points, providing a novel algebraic approach to understanding threshold risk sensitivity.

Findings

Finitely many risk critical points exist for polynomial and rational functions.

Risk critical points are characterized by zeroes of resolvent or leading coefficient of perturbed polynomials.

The equations for critical points are polynomial in the threshold parameter, enabling algebraic analysis.

Abstract

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 $\delta >0.$ 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 $\delta$ approaches these threshold values. We characterize candidates for risk critical points as zeroes of either the resolvent of a relevant $\delta-$perturbed polynomial, or of its leading coefficient, or both. Thus the equations that need to be solved are themselves polynomial equations in $\delta$ that exploit the algebraic properties of the underlying polynomial or rational functions. We name these important equations as "hidden equations of threshold risk".