Abstract The signature of a system is a probability vector that depends only on the system structure. Under the classic IID (independent and identically distributed) assumption on the component lifetimes, the system lifetime distribution is the convex combination of consecutive component failure times, and the signature coordinates constitute the mixture coefficients. In this case the signature representations are very useful in determining the system lifetime distributions and for stochastic comparisons of them. This first representation was obtained in 1985 by Samaniego. Then it was extended to the more general case of exchangeable component lifetimes. In 2011 Marichal, Mathonet and Waldhauser presented necessary and sufficient conditions assuring the Samaniego representation. There were expressed in terms of distributional properties of families of auxiliary indicator random vectors parametrized by positive numbers. In the paper we obtain other necessary and sufficient conditions represented in terms of the marginal distributions of component lifetimes and the dependence copula of them. Moreover, we study symmetry conditions for the equality of structural and probabilistic signatures.

中文翻译：

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用于系统生命周期签名表示的组件生命周期的边际和 copula 条件

摘要 系统的签名是一个仅依赖于系统结构的概率向量。在组件寿命的经典 IID（独立同分布）假设下，系统寿命分布是连续组件故障时间的凸组合，特征坐标构成混合系数。在这种情况下，签名表示在确定系统寿命分布和对它们进行随机比较时非常有用。Samaniego 于 1985 年获得了第一个表示。然后它扩展到可更换组件寿命的更一般情况。2011 年，Marichal、Mathonet 和 Waldhauser 提出了保证 Samaniego 代表性的充分必要条件。用正数参数化的辅助指标随机向量族的分布特性来表示。在本文中，我们获得了其他充分必要条件，这些条件以组件寿命的边际分布及其依赖关系表示。此外，我们研究了结构和概率签名相等的对称条件。