Nonparametric predictive inference for failure times of systems with exchangeable components

Abstract

Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. The theory of system signatures [1] provides a powerful framework for reliability assessment for systems consisting of exchangeable components. For a system with m components, the signature is a vector containing the probabilities for the events that the system fails at the moment of the j-th ordered component failure time, for all j = 1, . . . , m. As such, the signature represents the structure of the system. This paper presents how signatures can be used within nonparametric predictive inference, a statistical framework which uses few modelling assumptions enabled by the use of lower and upper probabilities to quantify uncertainty. The main result is the use of signatures to derive lower and upper survival functions for the failure time of systems with exchangeable components, given failure times of tested components that are exchangeable with those in the system. In addition, it is shown how the failure times of two such systems can be compared. This paper is the first in which signatures are combined with theory of lower and upper probabilities, related research challenges are briefly discussed.

Nonparametric predictive inference for system failure time based on bounds for the signature

“…Making this mistake, one could continue by calculating bounds for the full system's survival function following the standard way for simple parallel systems (effectively using '1 − (1 − S a )(1 − S b )', with self-explanatory notation). The resulting lower and upper survival functions are greater than (or equal to) the correctly derived bounds for the NPI lower and upper survival function, because for the correct method the dependence of the components in both systems is taken into account [5,6]. An intuitive explanation is as follows: The parallel system will only fail if both subsystems fail, and if one subsystem is known to fail this contains some information that suggests that the components are not very reliable, which as a consequence increases the (lower and upper) probability that the second subsystem also fails (when compared to the situation with the wrongly assumed independence between the two subsystems).…”

“…While this is a straightforward generalization of (1), the derivation involves m optimisation problems which take on the optima simultaneously [6]. For more information about the statistical framework of NPI, the theory of imprecise probability and applications in the area of reliability, the reader is referred to [5,9,10] 1 .…”

“…This reflects that there is no evidence in favour of such components, and hence the system, surviving past time t n (reflected by the lower survival function being zero), but the evidence against this is limited as there are only n observations (reflected by the upper survival function being a positive decreasing function of n). The NPI lower and upper survival functions for the failure time T S of a system with signature q are [6] …”

“…In NPI for system reliability, lower and upper survival functions are derived for the system's failure time, these reflect the limited knowledge about reliability of the components, using only the information from component tests. A brief introduction to NPI and overview of the results in [6] is given in Section 2.…”

“…Recently, the use of signatures for nonparametric predictive inference (NPI) for system reliability has been presented [6]. In NPI for system reliability, lower and upper survival functions are derived for the system's failure time, these reflect the limited knowledge about reliability of the components, using only the information from component tests.…”