Conditions on marginals and copula of component lifetimes for signature representation of system lifetime
Introduction
The general interest of the concept of signature in the field of reliability is a well-known fact. The study of such a concept leads to analyze interesting and specific aspects of stochastic dependence among non-negative random variables. More precisely, such a study leads to compare different properties of partial symmetry, which can be respectively seen as suitable generalizations of the exchangeability condition.
The first signature representation was obtained by Samaniego [22] for coherent systems with independent and identically (IID) distributed components having a common continuous distribution function F. This representation proves that the system distribution is a mixture (linear combination with non-negative weights) of the distributions of the ordered component lifetimes. The vector formed with the coefficients in that representation does not depend on F and was called the signature of the system. It is composed of the probabilities that the system fails with the consecutive component failures (for the formal definitions of probabilistic and structural signatures see Section 2). These distributions coincide with that of the order statistics and so this representation can be used to compute the system reliability from its signature. It can also be used to compare two systems with different structures just by comparing their signature vectors [7], [10], [15], [20].
This representation was extended in [14] to systems with component lifetimes having a joint absolutely continuous exchangeable (EXC) distribution. Exchangeable means that the distribution (law) is invariant under permutations. Then it was extended to general EXC distributions in [15] but in this case the signature values should be computed from the system structure and cannot be interpreted as probabilities. A review on the properties and applications of signature representations for systems and networks can be seen in [23]. Extensions to multi-state systems and connections with fuzzy measures were given in [17], [30].
Example 5.1 in [15] proves that the signature representations do not hold for systems with independent non-identically distributed components. We show here that the ID assumption cannot be relaxed as far as one needs conditions for the validity of the signature representation for any coherent system. Note that for k-out-of-n no conditions are required. However, two recent results prove that the EXC assumption can be relaxed.
The first one was obtained in Theorem 4 of [9]. There it is proved that a necessary and sufficient condition to get the signature representations of all the coherent systems is that the random variables with the components' states at time t are EXC for any .
The second one, obtained in Theorem 1 of [12], was based on the copula representation of the joint distribution of the component lifetimes. It is well known that this joint distribution is EXC if and only if the component (marginal) distributions are equal (ID) and the copula is EXC. It was proved in [12] that the EXC property for the copula can be relaxed. It is enough to merely assume that the copula is diagonal dependent (DD) (the formal definition is given in Section 2). Moreover, it was shown there that the set of DD copulas is much bigger than the set of EXC copulas and that it is dense in the set of copulas while the set of EXC copulas is not. Diagonal dependence of the copula is a sufficient condition on the dependence structure of component lifetimes for the signature representation of the system lifetime distribution, and this assumption is relaxed here so to obtain the necessary and sufficient condition.
In the present paper we obtain two new necessary and sufficient conditions for getting the signature representations for arbitrary systems. Other symmetry conditions are studied as well to get the equality between structural and probabilistic signatures. The rest of the paper is organized as follows. The main results are included in the following section. Examples and counterexamples are placed in Section 3. In Section 4, we analyze reliability models constructed with the use of multivariate conditional hazard rate functions, and investigate conditions on these functions which assure that the probabilistic and structural signatures of all the coherent systems are identical. A summary and some conclusions are given in Section 5.
Throughout the paper the terms ‘increasing’ and ‘decreasing’ are used in a wide sense, that is, they mean ‘non-decreasing’ and ‘non-increasing’, respectively. We use the notation and , , for a vector with coordinates when and otherwise.
Section snippets
Main results
A (binary) system is a Boolean (structure) function . Here means that the ith component does not work and that it works. Then the system state is completely determined by the structure function ψ and the component states . A system ψ is semi-coherent if it is increasing, and . A system is coherent if it is increasing and all the components are relevant. The ith component is relevant if ψ is strictly increasing in at
Examples
The first example shows that the ID condition is not enough for getting either of useful formulae (2.2), (2.4), (2.9), (2.11) or when the component lifetimes dependence structure does not fulfill appropriate symmetry (e.g., EXC) conditions.
Example 3.1 We treat the coherent system . Its structure signature is and its minimal and maximal signatures are and , respectively. We assume that the component lifetimes are ID with a common
Signatures in the uniform frailty model
In this section we consider the uniform frailty reliability model which assumes particular symmetry properties of multivariate conditional hazard rates. Our purpose is to show that the uniform frailty property implies identity of the probabilistic and structural signatures (2.1) and (2.3), respectively. First we recall the basic notions. In what follows we assume that has an absolutely continuous joint distribution.
Definition 4.1 For every , every elements of , and any
Summary and conclusion
This section is devoted to comment on main results of the paper and to put them in an historical perspective. Several interesting studies have been devoted, in the last decade, to conceptual aspects of the general concept of signature. However, only in the cases when the signature representation (2.4) holds, one gets a really efficient tool for reliability analysis.
Actually (2.4) can be seen as a condition of symmetry on the distribution of the components' lifetimes , which is implied by
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We would like to thank the anonymous AE and reviewers for several helpful suggestions that have served to improve this paper.
JN is partially supported by Ministerio de Ciencia e Innovación of Spain under grant PID2019-103971GB-I00, and FS by Sapienza Research Project Modelli stocastici con applicazioni nelle scienze e nell'ingegneria (2017) and Sapienza Research Project Simmetrie e Disuguaglianzein Modelli Stocastici (2018).
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