On modular decompositions of system signatures

Considering a semicoherent system made up of $n$ components having i.i.d. continuous lifetimes, Samaniego defined its structural signature as the $n$-tuple whose $k$-th coordinate is the probability that the $k$-th component failure causes the system to fail. This $n$-tuple, which depends only on the structure of the system and not on the distribution of the component lifetimes, is a very useful tool in the theoretical analysis of coherent systems. It was shown in two independent recent papers how the structural signature of a system partitioned into two disjoint modules can be computed from the signatures of these modules. In this work we consider the general case of a system partitioned into an arbitrary number of disjoint modules organized in an arbitrary way and we provide a general formula for the signature of the system in terms of the signatures of the modules. The concept of signature was recently extended to the general case of semicoherent systems whose components may have dependent lifetimes. The same definition for the $n$-tuple gives rise to the probability signature, which may depend on both the structure of the system and the probability distribution of the component lifetimes. In this general setting, we show how under a natural condition on the distribution of the lifetimes, the probability signature of the system can be expressed in terms of the probability signatures of the modules. We finally discuss a few situations where this condition holds in the non-i.i.d. and nonexchangeable cases and provide some applications of the main results

Construction of asymmetric copulas and its application in two-dimensional reliability modelling

Copulas offer a useful tool in modelling the dependence among random variables. In the literature, most of the existing copulas are symmetric while data collected from the real world may exhibit asymmetric nature. This necessitates developing asymmetric copulas that can model such data. In the meantime, existing methods of modelling two-dimensional reliability data are not able to capture the tail dependence that exists between the pair of age and usage, which are the two dimensions designated to describe product life. This paper proposes two new methods of constructing asymmetric copulas, discusses the properties of the new copulas, and applies the method to fit two-dimensional reliability data that are collected from the real world

Warranty Data Analysis: A Review

Warranty claims and supplementary data contain useful information about product quality and reliability. Analysing such data can therefore be of benefit to manufacturers in identifying early warnings of abnormalities in their products, providing useful information about failure modes to aid design modification, estimating product reliability for deciding on warranty policy and forecasting future warranty claims needed for preparing fiscal plans. In the last two decades, considerable research has been conducted in warranty data analysis (WDA) from several different perspectives. This article attempts to summarise and review the research and developments in WDA with emphasis on models, methods and applications. It concludes with a brief discussion on current practices and possible future trends in WDA

Susceptibility sets and the final outcome of collective Reed–Frost epidemics

This paper is concerned with exact results for the final outcome of stochastic SIR (susceptible → infective → recovered) epidemics among a closed, finite and homogeneously mixing population. The factorial moments of the number of initial susceptibles who ultimately avoid infection by such an epidemic are shown to be intimately related to the concept of a susceptibility set. This connection leads to simple, probabilistically illuminating proofs of exact results concerning the total size and severity of collective Reed–Frost epidemic processes, in terms of Gontcharoff polynomials, first obtained in a series of papers by Claude Lef`evre and Philippe Picard. The proofs extend easily to include general final state random variables defined on SIR epidemics, and also to multitype epidemics

A review of multi-component maintenance models with economic dependence

In this paper we review the literature on multi-component maintenance models with economic dependence. The emphasis is on papers that appeared after 1991, but there is an overlap with Section 2 of the most recent review paper by Cho and Parlar (1991). We distinguish between stationary models, where a long-term stable situation is assumed, and dynamic models, which can take information into account that becomes available only on the short term. Within the stationary models we choose a classification scheme that is primarily based on the various options of grouping maintenance activities: grouping either corrective or preventive maintenance, or combining preventive-maintenance actions with corrective actions. As such, this classification links up with the possibilities for grouped maintenance activities that exist in practice

An Introduction to Statistical Issues and Methods in Metrology for Physical Science and Engineering

This article provides an overview of the interplay between statistics and measurement. Measurement quality affects inference from data collected and analyzed using statistical methods while appropriate data analysis quantifies the quality of measurements. This article brings material on statistics and measurement together in one place as a resource for practitioners. Both frequentist and Bayesian methods are discussed

Trends in the Statistical Assessment of Reliability

Changes in technology have had and will continue to have a strong effect on changes in the area of statistical assessment of reliability data. These changes include higher levels of integration in electronics, improvements in measurement technology and the deployment of sensors and smart chips into more products, dramatically improved computing power and storage technology, and the development of new, powerful statistical methods for graphics, inference, and experimental design and reliability test planning. This paper traces some of the history of the development of statistical methods for reliability assessment and makes some predictions about the future