In this paper, we obtain ordering properties for coherent systems with possibly dependent identically distributed components. These results are based on a representation of the system reliability function as a distorted function of the common component reliability function. So, the results included in this paper can also be applied to general distorted distributions. The main advantage of these results is that they are distribution-free with respect to the common component distribution. Moreover, they can be applied to systems with component lifetimes having a non-exchangeable joint distribution. Appl. Stochastic ModelsBus. Ind. 2013, 29 264-278 J. NAVARRO ET AL.The rest of the paper is organized as follows. In Section 2, we give the main results of the paper with the general ordering results for distorted distributions and coherent systems with ID components. Specifically, we study properties of the usual stochastic order, hazard rate order, the reversed hazard rate order, the likelihood ratio order, the increasing convex order, the total time on test transform order, and the excess wealth order. In Section 3, we give some examples to show how to apply our general results to specific systems. These examples include systems with IID components and systems with dependent ID components. In Section 4, we give some conclusions and open questions for future research.Throughout the paper, we use the terms increasing and decreasing in a wide sense, that is, a function g is increasing (decreasing) if g.x/ 6 g.y/.>/ for all x 6 y. Whenever we consider an expectation (or a conditional random variable), we assume that it exists. The main resultsLet us consider a coherent system with lifetime T D .X 1 ; : : : ; X n / based on possibly dependent components with lifetimes X 1 ; : : : ; X n , where is the structure function (see [29], Chapter 1). Let us assume that X 1 ; : : : ; X n are identically distributed (ID) with a common reliability function F .t/ D Pr.X i > t/ for i D 1; : : : ; n. The component lifetimes X 1 ; : : : ; X n can be dependent, and this dependence will be represented by the joint reliability (or survival) function of .X

The preservation of reliability aging classes under the formation of coherent systems is a relevant topic in reliability theory. Thus, it is well known that the new better than used class is preserved under the formation of coherent systems with independent components. However, surprisingly, the increasing failure rate class is not preserved in the independent and identically distributed case, that is, the components may have the (negative) aging increasing failure rate property, but the system does not have this property. In this paper, we study conditions for the preservation of the main reliability classes under the formation of general coherent systems. These results can be applied both for systems with independent or dependent components. We consider both the case of systems with identically distributed components and the case of systems with components having different distributions. Copyright © 2013 John Wiley & Sons, Ltd.

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We prove that different conditional distributions can be represented as distorted distributions. These representations are used to obtain stochastic comparisons and bounds for them based on properties of the underlying copula. These properties can be used to explain the meaning of mathematical properties of copulas connecting them with dependence concepts. Some applications and illustrative examples are provided as well.