Shahid Chamran University of AhvazJournal of Advanced Mathematical Modeling2251-808810220201221Stochastic Comparison of $k$-out-of-$n$ Systems Based on Distortion FunctionStochastic Comparison of $k$-out-of-$n$ Systems Based on Distortion Function3563781554710.22055/jamm.2020.30655.1748FAElhamKhaleghpanah NoughabiDepartment of Statistics, University of Birjand, Birjand, IranMajidRezaeiDepartment of Statistics, University of Birjand, Birjand, IranMajidChahkandiDepartment of Statistics, University of Birjand, Birjand, IranJournal Article20190812One of the relevant problem in the reliability theory is stochastic comparison of coherent systems. Several results have been obtained in stochastic comparison of systems with independent and identically (IID) components. In this paper, we focus on $ k $-out-of-$ n $ systems that play an important role in study of the reliability of engineering systems. We obtain some results on distribution-free comparisons of $ k $-out-of-$ n $ systems, with possibly dependent component lifetimes, based on the concept of distortion function. We provide some conditions on distorted distributions of $ k $-out-of-$ n $ systems or their residual lifetimes that conclude ordering between their lifetimes or their residual lifetimes. As a special case we consider two common survival copula (Farlie-Gumbel-Morgenstern and Clayton–Oakes) to derive more details on stochastic comparison of $ k $-out-of-$ n $ systems with respect to $ k $ and $ n $. Some illustrative examples are also presented to show that some of results for stochastic comparison of $ k $-out-of-$ n $ systems with i.i.d components are violated in dependent case.One of the relevant problem in the reliability theory is stochastic comparison of coherent systems. Several results have been obtained in stochastic comparison of systems with independent and identically (IID) components. In this paper, we focus on $ k $-out-of-$ n $ systems that play an important role in study of the reliability of engineering systems. We obtain some results on distribution-free comparisons of $ k $-out-of-$ n $ systems, with possibly dependent component lifetimes, based on the concept of distortion function. We provide some conditions on distorted distributions of $ k $-out-of-$ n $ systems or their residual lifetimes that conclude ordering between their lifetimes or their residual lifetimes. As a special case we consider two common survival copula (Farlie-Gumbel-Morgenstern and Clayton–Oakes) to derive more details on stochastic comparison of $ k $-out-of-$ n $ systems with respect to $ k $ and $ n $. Some illustrative examples are also presented to show that some of results for stochastic comparison of $ k $-out-of-$ n $ systems with i.i.d components are violated in dependent case.https://jamm.scu.ac.ir/article_15547_fa37c8d3b4035b197e49bad299a530e4.pdf