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We consider an (n−k+1)-out-of-n concomitant system consisting of n components each having two subcomponents. This system functions if and only if at least (n−k+1) of the first subcomponents function, and the second subcomponents of working first components also function. The reliability of the proposed system is derived. The effect of dependent subcomponents on the system reliability relative to independent subcomponents is discussed. The system with two subcomponents is extended to the system with m subcomponents. Comparative numerical results and graphical representations are provided.

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... Therefore, they have applications in many fields such as selection procedures, inference problems, double sampling plans and systems reliability. For example, in [3,4] are studied from a reliability point of view complex systems with components which have two subcomponents that performs different tasks, and in [5], the distribution theory of lifetimes of two component systems is discussed. In studies regarding the concomitants there are two elements that have to be mentioned: the kind of dependence between first and second variate, and the kind of order for the first variate. ...

... In this paper, we focus on the concomitants of GOS and with the dependence structure between the first variate and the second variate given by the Fairlie-Gumbel-Morgenstern (FGM) family. This family is a flexible family of bivariate distributions used as a modeling tool for bivariate data in many fields [7], one such field being Reliability, see [3][4][5]. The FGM family has a simple analytical form, but it can describe only relatively weak dependence because the correlation coefficient between the two components cannot exceed 1/3. ...

... Then, the corresponding Y-values represent performance on an associated characteristics. In Reliability Theory, the role of the concomitants is emphasized in [3][4][5]. ...

In this paper we recall, extend and compute some information measures for the concomitants of the generalized order statistics (GOS) from the Farlie–Gumbel–Morgenstern (FGM) family. We focus on two types of information measures: some related to Shannon entropy, and some related to Tsallis entropy. Among the information measures considered are residual and past entropies which are important in a reliability context.

In this article, we introduce a new family of mixed shock models. Consider a system whose efficiency is k and the system is affected by r types of shocks, when the efficiency of the system reduces by one unit, regarding the occurrence of each one of the shocks. If K1 shocks of type I, K2 shocks of type II, …, Kr shocks of type r occur, in which K1+K2+…+Kr=k, the system loses its efficiency and fails. The system lifetime reliability function, the reliability of system lifetime in a perfectly functioning state, the reliability of system lifetime in partially working states, and the Laplace transformation of system lifetime have been calculated. In the following, the effect of critical point values on the reliability function of the system lifetime has been studied by using the simulation.

Pareto distributions are very flexible probability models with various forms and kinds. In this paper, a new bivariate Pseudo-Pareto distribution and its properties are presented and discussed. Main variables, order statistics and concomitants of this distribution are studied and their importance for risk and reliability analysis is explained. Joint and marginal distributions, complementing cumulative distributions and hazard functions of the variables are derived. Numerical illustrations, graphical displays and interpretations for the obtained distributions and derived functions are provided. An implementation example on defaultable bonds is performed.

The number of failed components in a failed or operating system is a very useful quantity in terms of replacement and maintenance strategies. These quantities have been studied in several papers for a system consisting of identical components. In this paper, the number of failed components at the time when the system fails and the number of failed components when the system is working are considered for a well-known and widely applicable k-out-of-n structure. The system is assumed to have multiple types of components. That is, the system consists of components having nonidentical failure time distributions. Optimization problems are also formulated to find optimal values of the number of components of each type, and the optimal replacement time.

Mean residual life is a useful dynamic characteristic to study reliability of a system. It has been widely considered in the literature not only for single unit systems but also for coherent systems. This article is concerned with the study of mean residual life for a coherent system that consists of multiple types of dependent components. In particular, the survival signature based generalized mixture representation is obtained for the survival function of a coherent system and it is used to evaluate the mean residual life function. Furthermore, two mean residual life functions under different conditional events on components’ lifetimes are also defined and studied.

The concept of the survival signature has recently attracted increasing attention for performing reliability analysis on systems with multiple types of components. It opens a new pathway for a structured approach with high computational efficiency based on a complete probabilistic description of the system. In practical applications, however, some of the parameters of the system might not be defined completely due to limited data, which implies the need to take imprecisions of component specifications into account. This paper presents a methodology to include explicitly the imprecision, which leads to upper and lower bounds of the survival function of the system. In addition, the approach introduces novel and efficient component importance measures. By implementing relative importance index of each component without or with imprecision, the most critical component in the system can be identified depending on the service time of the system. Simulation method based on survival signature is introduced to deal with imprecision within components, which is precise and efficient. Numerical example is presented to show the applicability of the approach for systems.

Suppose the independent pairs of variates (Xi, Yi), i = l,…,n, are ordered by the Xi. Then the Y-variate paired with the r-th order statistic Xr:n is called the concomitant of Xr:n and denoted by Y[r:n]. This paper treats the asymptotic distribution theory of concomitants when r or n-r remains fixed as n→∞. The special case of a linear regression linking X and Y is examined in detail and a theorem of Galambos (1978) is generalized. Results on the joint distribution of concomitants of extremes are reviewed and some applications are indicated.

A coherent system of order n that consists two different types of dependent components is considered. The lifetimes of the components in each group are assumed to follow an exchangeable joint distribution, and the two random vectors, which represent the lifetimes of the components in each group are also assumed to be dependent. Under this particular form of dependence, all components are assumed to be dependent but they are categorized with respect to their reliability functions. Mixture representation is obtained for the survival function of the system's lifetime. Mixture representations are also obtained for the series and parallel systems consisting of disjoint modules such that all components of Type I are involved in one module (subsystem) and all components of Type II are placed in the other module. The theoretical results are illustrated with examples. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 388–394, 2015

Nonparametric predictive inference for system reliability has recently been presented, with specific focus on k -out-of-m :G systems. The reliability of systems is quantified by lower and upper probabilities of system functioning, given binary test results on components, taking uncertainty about component functioning and indeterminacy due to limited test information explicitly into account. Thus far, systems considered were series configurations of subsystems, with each subsystem i a k i-out-of-mi:Gmi:G system which consisted of only one type of components. Key results are briefly summarized in this paper, and as an important generalization new results are presented for a single k-out-of-m:G system consisting of components of multiple types. The important aspects of redundancy and diversity for such systems are discussed.

The reliability and mean residual life of complex systems are discussed. These systems consist of n elements each having two s -dependent subcomponents. The reliability of such systems involves the distributions of bivariate order statistics, and are connected with a bivariate binomial distribution. The mean residual life function of complex systems with intact components at time t is also discussed. Some examples and graphical representations are given.

Let X1:n≤X2:n⋯≤Xn:nX1:n≤X2:n⋯≤Xn:n be the order statistics from some sample, and let Y[1:n],Y[2:n],…,Y[n:n]Y[1:n],Y[2:n],…,Y[n:n] be the corresponding concomitants. One purpose of this paper is to obtain results that stochastically compare, in various senses, the random vector (Xr:n,Y[r:n])(Xr:n,Y[r:n]) to the random vector (Xr+1:n,Y[r+1:n])(Xr+1:n,Y[r+1:n]), r=1,2,…,n−1r=1,2,…,n−1. Such comparisons are called one-sample comparisons. Next, let S1:n≤S2:n⋯≤Sn:nS1:n≤S2:n⋯≤Sn:n be the order statistics constructed from another sample, and let T[1:n],T[2:n],…,T[n:n]T[1:n],T[2:n],…,T[n:n] be the corresponding concomitants. Another purpose of this paper is to obtain results that stochastically compare, in various senses, the random vector (Xr:n,Y[r:n])(Xr:n,Y[r:n]) with the random vector (Sr:n,T[r:n])(Sr:n,T[r:n]), r=1,2,…,nr=1,2,…,n. Such comparisons are called two-sample comparisons. It is shown that some of the results in this paper strengthen previous results in the literature. Some applications in reliability theory are described.

In this paper, we study how to compute the signature of a k-out-of-n coherent system consisting of n modules. Formulas for computing the signature and the minimal signature of this kind of systems based on those of their modules are derived. Examples are presented to demonstrate the applications of our formulas. The main results obtained in this paper generalize some related ones in recent literature.

This chapter discusses the theory and applications related to the induced order statistics. Recently a systematic study of the induced order statistics, their ranks, their extremes and their partial sums are undertaken. The chapter assumes that X and Y are two numerical characteristics defined for each individual in a population. Induced order statistics arise naturally in the context of selection where individuals ought to be selected by their ranks in respect of Y, but are selected by their ranks in a related variate X due to unavailability of Y at the time of selection. Induced order statistics are also useful in regression analysis, especially when the observations are subject to a type II censoring scheme with respect to the dependent variable, or when the regression function at a given quantile of the predictor variable is of interest.

Concomitants of order statistics are considered for the situation in which the random vectors (X1, Y1), (X2, Y2),…, (Xn, Yn) are independent but otherwise arbitrarily distributed. The joint and marginal distributions of the concomitants of order statistics and stochastic comparisons among the concomitants of order statistics are studied in this situation.

For a random sample of size n from an absolutely continuous random vector (X,Y), let Yi:n be ith Y-order statistic and Y[j:n] be the Y-concomitant of Xj:n. We determine the joint pdf of Yi:n and Y[j:n] for all i,j=1 to n, and establish some symmetry properties of the joint distribution for symmetric populations. We discuss the uses of the joint distribution in the computation of moments and probabilities of various ranks for Y[j:n]. We also show how our results can be used to determine the expected cost of mismatch in broken bivariate samples and approximate the first two moments of the ratios of linear functions of Yi:n and Y[j:n]. For the bivariate normal case, we compute the expectations of the product of Yi:n and Y[i:n] for n=2 to 8 for selected values of the correlation coefficient and illustrate their uses.

We consider a generalization of the bivariate Farlie-Gumbel-Morgenstern (FGM) distribution by introducing additional parameters. For the generalized FGM distribution, the admissible range of the association parameter allowing positive quadrant dependence property is shown. Distributional properties of concomitants for this generalized FGM distribution are studied. Recurrence relations between moments of concomitants are presented.

We study the distributions of Y-concomitants of the X-order statistics for a special dependent sample (Xi,Yi), i=1,...,n. The dependence among the sample is due to the Xi's, which are assumed to be distributed as equally-correlated multivariate normal. The finite-sample and asymptotic distributions of concomitants are derived under this setup.