In this paper, we introduce a novel way to quantify the remaining inaccuracy of order statistics by utilizing the concept of extropy. We explore various properties and characteristics of this new measure. Additionally, we expand the notion of inaccuracy for ordered random variables to a dynamic version and demonstrate that this dynamic information measure provides a unique determination of the distribution function. Moreover, we investigate specific lifetime distributions by analyzing the residual inaccuracy of the first-order statistics. Nonparametric kernel estimation of the proposed measure is suggested. Simulation results show that the kernel estimator with bandwidth selection using the cross-validation method has the best performance. Finally, an application of the proposed measure on the model selection is provided.

This paper examines the preservation of several aging classes of lifetime distributions in the formation of coherent and mixed systems with independent and identically distributed (i.i.d.) or identically distributed (i.d.) component lifetimes. The increasing mean inactivity time class and the decreasing mean time to failure class are developed for the lifetime of systems with possibly dependent and i.d. component lifetimes. The increasing likelihood ratio property is also discussed for the lifetime of a coherent system with i.i.d. component lifetimes. We present sufficient conditions satisfied by the signature of a coherent system with i.i.d. components with exponential distribution, under which the decreasing mean remaining lifetime, the increasing mean inactivity time, and the decreasing mean time to failure are all satisfied by the lifetime of the system. Illustrative examples are presented to support the established results.

This paper discusses stochastic comparisons for the residual and past lifetimes of coherent systems with dependent and identically distributed (d.i.d.) components under random monitoring in terms of the hazard rate, the reversed hazard rate, and the likelihood ratio orders. Some stochastic comparisons results are also established on the residual lifetimes of coherent systems under random observation times when all of the components are alive at that time. Sufficient conditions are established in terms of the aging properties of the components and the distortion functions induced from the system structure and dependence among components lifetimes. Numerical examples are provided to illustrate the theoretical results as well.

Relative ageing describes how one system ages with respect to another. The ageing faster orders are used to compare the relative ageing of two systems. Here, we study ageing faster orders in the hazard and reversed hazard rates. We provide some sufficient conditions for one coherent system to dominate another with respect to ageing faster orders. Further, we investigate whether the active redundancy at the component level is more effective than that at the system level with respect to ageing faster orders, for a coherent system. Furthermore, a used coherent system and a coherent system made out of used components are compared with respect to ageing faster orders.

The aim of this paper is to introduce a new stochastic order based on the residual lifetimes of two nonnegative dependent random variables and the stochastic precedence order. We develop some characterizations and preservation properties of this stochastic order. In addition, we study some of its reliability properties and its relation with other existing stochastic orders. One of the possible applications in reliability theory has also been discussed.

Sequential order statistics can be used to describe the ordered lifetimes of components of a system when the failure of a component may affect the reliability of the remaining components. After a reliability system consisting of n components fails, some of its components may still be alive. In this paper we first establish some univariate stochastic orderings and ageing properties of the residual lifetimes of the live components in a sequential (n-r+1)-out-of-n system. We also obtain a characterizing result for the exponential distribution based on uncorrelated residual lifetimes of live components. Finally, we provide some sufficient conditions for comparing vectors of residual lifetimes of the live components from two sequential (n-r+1)-out-of-n systems. The results established here extend some well-known results in the literature.

In this paper we derive mixture representations for the reliability functions of the conditional residual life and inactivity time of a coherent system with n independent and identically distributed components. Based on these mixture representations we carry out stochastic comparisons on the conditional residual life, and the inactivity time of two coherent systems with independent and identical components.

This paper builds a mixture representation of the reliability function of the conditional residual lifetime of a coherent system in terms of the reliability functions of conditional residual lifetimes of order statistics. Some stochastic ordering properties for the conditional residual lifetime of a coherent system with independent and identically distributed components are obtained, based on the stochastically ordered coefficient vectors.

In the study of the reliability of technical systems in reliability engineering, coherent systems play a key role. In this paper we consider a coherent system consisting of n components with independent and identically distributed components and propose two time-dependent criteria. The first criterion is a measure of the residual lifetime of live components of a coherent system having some of the components alive when the system fails at time t. The second criterion is a time-dependent measure which enables us to investigate the inactivity times of the failed components of a coherent system still functioning though some of its components have failed. Several ageing and stochastic properties of the proposed measures are then established.

The conditional tail expectation in risk analysis describes the expected amount of risk that can be experienced given that a potential risk exceeds a threshold value, and provides an important measure of right-tail risk. In this paper, we study the convolution and extreme values of dependent risks that follow a multivariate phase-type distribution, and derive explicit formulae for several conditional tail expectations of the convolution and extreme values for such dependent risks. Utilizing the underlying Markovian property of these distributions, our method not only provides structural insight, but also yields some new distributional properties of multivariate phase-type distributions.

Recently Bassan and Spizzichino (1999) have given some new concepts of multivariate ageing for exchangeable random variables, such as a special type of bivariate IFR, by comparing distributions of residual lifetimes of dependent components of different ages. In the same vein, we further study some properties of these concepts of IFR in the bivariate case. Then we introduce certain concepts of bivariate DMRL ageing and we develop a treatment that parallels those developed for bivariate IFR. For both the IFR and DMRL concepts, we analyse a weak and a strong version, and discuss some of the differences between them.

As proposed by Ebrahimi, uncertainty in the residual lifetime distribution can be measured by means of the Shannon entropy. In this paper, we analyse a dual characterization of life distributions that is based on entropy applied to the past lifetime. Various aspects of this measure of uncertainty are considered, including its connection with the residual entropy, the relation between its increasing nature and the DRFR property, and the effect of monotonic transformations on it.

For a class of renewal queueing processes characterized by a rational Laplace–Stieltjes transform of the arrival inter-occurrence time distribution, the Laplace–Stieltjes transform of the equilibrium (actual) waiting time distribution is re-expressed in a manner which facilitates explicit inversion under certain conditions. The results are of interest in other contexts as well, as for example in insurance ruin theory. Various analytic properties of these quantities are then obtained as a result.

This paper establishes the existence of the limit as t → ∞ of the distribution of the residual lifetime at time t of a two-unit warm-standby reliability system supported by a single repair facility, conditioned on the event that the system has not been down at any time in (0, t). The conditioned limit distribution is proved to be exponential irrespective of the distributions of the lifetimes and repair times of the individual units, provided that their Laplace transforms are rational functions of their arguments.