Exact Critical Power Indices for Heterogeneous PRHR Series Systems Under Archimedean Dependence
- 1Department of Quantitative Analysis, College of Business Administration, King Saud University, Riyadh 11362, Saudi Arabia
- 2Department of Statistics and Operations Research, College of Science, King Saud University, Riyadh 11451, Saudi Arabia
Abstract
For a series system with heterogeneous proportional reversed-hazard-rate components coupled through an Archimedean survival copula, we determine the exact admissible range of power-mean indices for a universal homogeneous stochastic upper envelope. The characterization reduces to the convexity of a single transformed inverse-generator function; under complete monotonicity, this condition is necessary already from the two-component case. The resulting critical power index is sharp: every order at or above the threshold is universally valid, whereas every smaller order fails for a suitable two-component perturbation. Under independence, the threshold is zero, making the geometric mean optimal. Within the Clayton family, negative critical indices occur, with an exact threshold −a for 0 < a ≤ 1 and a threshold strictly below −1 for a > 1. Under additional generator concavity and monotonicity conditions, the same construction yields the stronger reversed hazard rate order, with exact thresholds for independence and for the Clayton model at a = 1. To address dependence uncertainty, we also derive a sharp classwise robust threshold and show that a negative Clayton-optimal order need not remain valid under an alternative Archimedean family. A deterministic sensitivity study confirms the computational scalability of the construction across system size, heterogeneity, dependence strength, and baseline choice.
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BibTeX
@article{alshehri2026exact,
title = {{Exact Critical Power Indices for Heterogeneous PRHR Series Systems Under Archimedean Dependence}},
author = {Mashael A. Alshehri and Maha Al Mutairi},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14193603},
url = {https://www.mdpi.com/2227-7390/14/19/3603},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Exact Critical Power Indices for Heterogeneous PRHR Series Systems Under Archimedean Dependence
AU - Mashael A. Alshehri
AU - Maha Al Mutairi
PY - 2026
DA - 2026/10/04
JO - Mathematics
DO - 10.3390/math14193603
UR - https://www.mdpi.com/2227-7390/14/19/3603
AB - For a series system with heterogeneous proportional reversed-hazard-rate components coupled through an Archimedean survival copula, we determine the exact admissible range of power-mean indices for a universal homogeneous stochastic upper envelope. The characterization reduces to the convexity of a single transformed inverse-generator function; under complete monotonicity, this condition is necessary already from the two-component case. The resulting critical power index is sharp: every order at or above the threshold is universally valid, whereas every smaller order fails for a suitable two-component perturbation. Under independence, the threshold is zero, making the geometric mean optimal. Within the Clayton family, negative critical indices occur, with an exact threshold −a for 0 < a ≤ 1 and a threshold strictly below −1 for a > 1. Under additional generator concavity and monotonicity conditions, the same construction yields the stronger reversed hazard rate order, with exact thresholds for independence and for the Clayton model at a = 1. To address dependence uncertainty, we also derive a sharp classwise robust threshold and show that a negative Clayton-optimal order need not remain valid under an alternative Archimedean family. A deterministic sensitivity study confirms the computational scalability of the construction across system size, heterogeneity, dependence strength, and baseline choice.
SN - 2227-7390
ER -