## 1 Introduction

The study on ordering properties of coherent systems is a relevant subject in reliability and survival theories. The classical signature-based mixture representation obtained by Samaniego [1] (see also [2]) for systems with independent and identically distributed (IID) components was used by Kochar *et al.* [3] to obtain some stochastic ordering properties. These ordering properties are distribution-free; that is, they do not depend on the common component distribution. Samaniego's representation was extended by Navarro *et al.* [4] to the case of systems with exchangeable components, that is, systems with component lifetimes having a joint distribution invariant under permutations. Moreover, they obtain distribution-free ordering properties used to compare systems having different numbers of exchangeable components. Other ordering properties and bounds for systems were obtained in [5-14]. Unfortunately, Samaniego's representation does not necessarily hold when the components are not exchangeable (see Example 5.1 in [4]). In this case, some approximations are obtained in [15], and some particular ordering results have been obtained for specific coherent systems and component distributions (see, e.g., [16-19]).

In this paper, we study ordering properties of coherent systems with identically distributed (ID) components. Note that this case is more general than the exchangeable case. To this purpose, we obtain a representation of the system reliability as a *distorted function* of the common component reliability ; that is, , where *h* is an increasing function which depends on the system structure and the survival copula of the joint distribution of the component lifetimes. Then, we obtain ordering properties for distorted distributions and use these results to obtain ordering properties for coherent systems. As we use a different approach to that used in [4], these results also have interest for systems with exchangeable (or IID) components. Moreover, we study some stochastic orders not studied in [4]. Our results can also be applied to study general distorted distributions in other fields. Distorted functions have been the subject of a wider interest because of their use in the rank-dependent expected utility model (see [20, 21]). Denneberg [22] and Wang [23, 24] introduced distorted distributions in insurance pricing and financial risk managements. Wang and Young [25] connected these types of distributions with stochastic orders. Recently, Sordo and Suárez-Llorens [26] have obtained some new results about stochastic orderings of distorted distributions. Also, some ordering results were obtained in Section 3.2 of Khaledi and Shaked's work [27].

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*) *≤ g*(*y*)(*≥*) for all *x ≤ y*. Whenever we consider an expectation (or a conditional random variable), we assume that it exists.