Use of copula functions in error assessment due to deviation from dependence assumption

TL;DR

This paper develops a copula-based framework to quantify relative errors incurred when independence is incorrectly assumed for lifetimes in $n$-component series and parallel reliability systems. By expressing survival and aging functions through copulas and their survival counterparts, it derives OA/UA behaviors and stochastic-order relations across a wide class of parametric copulas (including FGMs, Fischer-Köck, Gumbel-Hougaard, AMH, and Marshall-Olkin). It then applies these results to numerous multivariate exponential and Weibull models, detailing the corresponding reliability measures and their errors $E^{\overline{F}}$, $E^{r}$, $E^{\mu}$, and $E^{L}$, and showing how parameter choices drive the direction and magnitude of bias. The work also identifies discrepancies with prior literature (notably for Gumbel-I/II) and argues for a unified, copula-driven approach to error assessment with practical implications for reliability evaluation under dependence. Overall, the results provide a general toolkit for assessing when independence-based calculations are safely used and when dependence-aware corrections are warranted.

Abstract

In this paper, we analyze the relative errors in various reliability measures due to the tacit assumption that the components associated with a $n$-component series system or a parallel system are independently working where the components are dependent. We use Copula functions in said error analysis. This technique generalizes the existing work on error assessment for many wide class of distributions.

Theorems & Definitions (56)

• Theorem 2.1
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Proposition 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 3.1
• Theorem 3.2