Resolving an old problem on the preservation of the IFR property under the formation of -out-of- systems with discrete distributions

Abstract More than half a century ago, it was proved that the increasing failure rate (IFR) property is preserved under the formation of k -out-of- n systems (order statistics) when the lifetimes of the components are independent and have a common absolutely continuous distribution function. However, this property has not yet been proved in the discrete case. Here we give a proof based on the log-concavity property of the function $f({{\mathrm{e}}}^x)$ . Furthermore, we extend this property to general distribution functions and general coherent systems under some conditions.