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Use of stochastic orders and statistical dependence in error analysis for multi-component system

Subarna Bhattacharjee, Aninda Kumar Nanda, Subhashree Patra

TL;DR

This work analyzes the incurred error in reliability measures when replacing dependent lifetimes with independent ones in multi-component series and parallel systems, focusing on multivariate exponential and Weibull models. By leveraging statistical dependence concepts and stochastic orders, the authors derive explicit relative-error expressions for a wide range of multivariate distributions (MOME, MGI, MOMW, MCW, ML, FGMW) across both series and parallel configurations, and establish sign patterns and aging implications. They connect error behavior to orders such as ST, FR, MRL, and AI, showing how dependence can lead to over- or under-estimation of survival, failure rate, and aging metrics, with practical guidance on when simple independence-based analyses may be acceptable. The paper also offers comparative analyses and a comprehensive notation framework, enabling practitioners to gauge error bounds and select appropriate multivariate dependence structures in reliability assessment. Overall, the results provide a rigorous link between dependence, stochastic orderings, and relative errors, enriching reliability theory with quantitative dependence-aware guidance for multi-component systems.

Abstract

In this paper, we analyze the relative errors that crop up in the various reliability measures due to the tacit assumption that the components are independently working associated with a $n$-component series system or a parallel system where the components are dependent and follow a well-defined multivariate Weibull or exponential distribution. We also list some important observations which the previous authors have not noted in their earlier works. In this paper, we focus on the incurred error in multi-component series and parallel systems having multivariate Weibull distributions. In the upcoming sections, we establish that the present study has relevance with stochastic orders and statistical dependence which were not previously pointed out by previous authors.

Use of stochastic orders and statistical dependence in error analysis for multi-component system

TL;DR

This work analyzes the incurred error in reliability measures when replacing dependent lifetimes with independent ones in multi-component series and parallel systems, focusing on multivariate exponential and Weibull models. By leveraging statistical dependence concepts and stochastic orders, the authors derive explicit relative-error expressions for a wide range of multivariate distributions (MOME, MGI, MOMW, MCW, ML, FGMW) across both series and parallel configurations, and establish sign patterns and aging implications. They connect error behavior to orders such as ST, FR, MRL, and AI, showing how dependence can lead to over- or under-estimation of survival, failure rate, and aging metrics, with practical guidance on when simple independence-based analyses may be acceptable. The paper also offers comparative analyses and a comprehensive notation framework, enabling practitioners to gauge error bounds and select appropriate multivariate dependence structures in reliability assessment. Overall, the results provide a rigorous link between dependence, stochastic orderings, and relative errors, enriching reliability theory with quantitative dependence-aware guidance for multi-component systems.

Abstract

In this paper, we analyze the relative errors that crop up in the various reliability measures due to the tacit assumption that the components are independently working associated with a -component series system or a parallel system where the components are dependent and follow a well-defined multivariate Weibull or exponential distribution. We also list some important observations which the previous authors have not noted in their earlier works. In this paper, we focus on the incurred error in multi-component series and parallel systems having multivariate Weibull distributions. In the upcoming sections, we establish that the present study has relevance with stochastic orders and statistical dependence which were not previously pointed out by previous authors.

Paper Structure

This paper contains 50 sections, 10 theorems, 164 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Role of statistical dependence and stochastic orders in error analysis of reliability functions
  3. Use of Statistical dependence in error analysis
  4. Role of stochastic orders with an important note on existing literature
  5. Error Analysis of multi-component series systems
  6. Series system with components having multivariate Exponential distribution
  7. Independent Exponential
  8. Marshall and Olkin's Multivariate Exponential distribution ($MOME$)
  9. Multivariate Gumbel's Type I ($MGI$)
  10. Series system with components having Multivariate Weibull distribution
  11. Independent Weibull
  12. Marshal and Olkin (1967a)($MOMW$)
  13. Multivariate Crowder Weibull (1989) ($MCW$)
  14. Multivariate Lee (1979) ($ML$)
  15. Lu and Bhattcharyya (1990): I
  16. ...and 35 more sections

Key Result

Proposition 2.1

Let $T_1, T_2,\ldots,T_n$ denote the component lifetimes of a series (parallel) system and let relative error in $SF$ due to faulty assumption of components independently working in the series (parallel) system be denoted by $E^{\overline{F}}_{S}(t)$ ($E^{\overline{F}}_{P}(t)$). Then

Theorems & Definitions (17)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • Definition 2.3
  • Proposition 2.2
  • Definition 2.4
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • Corollary 2.1
  • ...and 7 more