Hidden equations of risk critical thresholds. Issue 2 (3rd April 2023)

• Record Type: Journal Article
• Authors: Ejov, Vladimir V.; Filar, Jerzy A.; Qiao, Zhihao
• Is Part Of: Stochastic models. Volume 39:Issue 2(2023)
• Page Start: 383
• Page End: 413
• Publication Date: 2023-04-03
• Subjects: Discriminant and Puiseux series -- polynomial perturbations -- roots of polynomials -- tail probabilities -- threshold risk; Stochastic processes -- Periodicals; Probabilities -- Periodicals; 519.2
• DOI: 10.1080/15326349.2022.2108452
• Languages: English
• ISSNs: 1532-6349

Abstract

Abstract: We consider the problem of parametric sensitivity of a particular characterization of risk, with respect to a threshold parameter δ . Such threshold risk is modeled as the probability of a δ − perturbed function of a random variable falling below 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 them. Under weak conditions, we characterize candidates for risk critical points as zeroes of either the discriminant 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 risk critical thresholds".