Stochastic Comparison of ‏‎$‎‎‎k‎$‎-out-of-‎‏‎$‎‎‎n‎$‎ Systems‎‏ ‎Based ‎on‎ Distortion Function

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.