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Stochastic comparison of series and parallel systems lifetime in Archimedean copula under random shock

Sarikul Islam, Nitin Gupta

TL;DR

This work addresses the problem of comparing lifetimes of series and parallel reliability systems whose components are dependent and heterogeneously distributed, subject to random shocks and described by distinct Archimedean copula structures. The authors develop sufficient conditions based on majorization, Schur-convexity, and the additive properties of the Copula composition $\psi_2\circ\phi_1$ to establish usual stochastic-order results, considering both increasing log-concave and decreasing log-convex survival functions. The main contributions include propositions and theorems that extend prior results to allow different dependency structures across systems and provide concrete copula examples (e.g., Clayton, AMH, Gumbel) with graphical demonstrations. These results have practical implications for reliability assessment, risk management, and actuarial settings where random shocks and component dependencies influence system lifetimes.

Abstract

In this paper, we studied the stochastic ordering behavior of series as well as parallel systems' lifetimes comprising dependent and heterogeneous components, experiencing random shocks, and exhibiting distinct dependency structures. We establish certain conditions on the lifetime of individual components where the dependency among components defined by Archimedean copulas, and the impact of random shocks on the overall system lifetime to get the results. We consider components whose survival functions are either increasing log-concave or decreasing log-convex functions of the parameters involved. These conditions make it possible to compare the lifetimes of two systems using the usual stochastic order framework. Additionally, we provide examples and graphical representations to elucidate our theoretical findings.

Stochastic comparison of series and parallel systems lifetime in Archimedean copula under random shock

TL;DR

This work addresses the problem of comparing lifetimes of series and parallel reliability systems whose components are dependent and heterogeneously distributed, subject to random shocks and described by distinct Archimedean copula structures. The authors develop sufficient conditions based on majorization, Schur-convexity, and the additive properties of the Copula composition to establish usual stochastic-order results, considering both increasing log-concave and decreasing log-convex survival functions. The main contributions include propositions and theorems that extend prior results to allow different dependency structures across systems and provide concrete copula examples (e.g., Clayton, AMH, Gumbel) with graphical demonstrations. These results have practical implications for reliability assessment, risk management, and actuarial settings where random shocks and component dependencies influence system lifetimes.

Abstract

In this paper, we studied the stochastic ordering behavior of series as well as parallel systems' lifetimes comprising dependent and heterogeneous components, experiencing random shocks, and exhibiting distinct dependency structures. We establish certain conditions on the lifetime of individual components where the dependency among components defined by Archimedean copulas, and the impact of random shocks on the overall system lifetime to get the results. We consider components whose survival functions are either increasing log-concave or decreasing log-convex functions of the parameters involved. These conditions make it possible to compare the lifetimes of two systems using the usual stochastic order framework. Additionally, we provide examples and graphical representations to elucidate our theoretical findings.
Paper Structure (6 sections, 14 theorems, 43 equations, 4 figures)

This paper contains 6 sections, 14 theorems, 43 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results for the super-additive case
  4. Main results for the sub-additive case
  5. Some applications
  6. Conclusion

Key Result

Lemma 2.1

(marshall2011preservation, Theorem 5.A.2) Consider the function h, which is increasing, and concave (convex), then $(x_1,\cdots,\,x_n) \preceq^w (\preceq_w) (y_1,\cdots,\,y_n) \implies(h(x_1),\cdots,\,h(x_n)) \preceq^w (\preceq_w) (h(y_1),\cdots,\,h(y_n)$ for $\bm{x},\,\bm{y} \in\mathbb{R}^n$.

Figures (4)

  • Figure 1: Graph of $F_{Y_{n:n}^p}(x)-F_{X_{n:n}^p}(x)$ for $\theta_1=0.2$ and $\theta_2=0.75$.
  • Figure 2: Graph of $F_{Y_{n:n}^p}(x)-F_{X_{n:n}^p}(x)$ for $\theta_1=0.6$ and $\theta_2=0.3$.
  • Figure 3: Graph of $F_{Y_{n:n}^p}(x) - F_{X_{n:n}^p}(x)$ for $\theta_1=3$ and $\theta_2=15$.
  • Figure 4: Graph of $F_{Y_{n:n}^p}(x) - F_{X_{n:n}^p}(x)$ for $\theta_1=9$ and $\theta_2=2$.

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Proposition 3.1
  • proof
  • ...and 29 more