Preservation of stochastic orders through the system signature has captured the attention of researchers in recent years. Signature-based comparisons have been made for the usual stochastic order, hazard rate order, and likelihood ratio orders. However, for the mean residual life (MRL) order, it has recently been proved that the preservation result does not hold true in general, but rather holds for a particular class of distributions. In this paper, we study whether or not a similar preservation result holds for the mean inactivity time (MIT) order. We prove that the MIT order is not preserved from signatures to system lifetimes with independent and identically distributed (i.i.d.) components, but holds for special classes of distributions. The relationship between these classes and the order statistics is also highlighted. Furthermore, the distribution-free comparison of the performance of coherent systems with dependent and identically distributed (d.i.d.) components is studied under the MIT ordering, using diagonal-dependent copulas and distorted distributions.

This paper investigates the issue of stochastic comparison of multi-active redundancies at the component level versus the system level. Based on the assumption that all components are statistically dependent, in the case of complete matching and nonmatching spares, we present some interesting comparison results in the sense of the hazard rate, reversed hazard rate and likelihood ratio orders, respectively. And we also obtain two comparison results between relative agings of resulting systems at the component level and the system level. Several numerical examples are provided to illustrate the theoretical results.

In this paper we investigate different methods that may be used to compare coherent systems having heterogeneous components. We consider both the case of systems with independent components and the case of systems with dependent components. In the first case, the comparisons are based on the new concept of the survival signature due to Coolen and Coolen-Maturi (2012) which extends the well-known concept of system signatures to the case of components with lifetimes that need not be independent and identically distributed. In the second case, the comparisons are based on the concept of distortion functions. A graphical procedure (called an RR-plot) is proposed as an alternative to the analytical methods when there are two types of components.