Stochastic comparison of lifetimes of two -out-of- systems with heterogeneous dependent components

In this paper, we shall generalize stochastic comparison of lifetimes of two ( n − k + 1 ) -out-of- n systems of possibly dependent lifetimes. The type of dependency assumed throughout this paper is according to Archimedean copulas with n -monotone and completely monotone (cm) generators. In fact, we provide certain conditions under which one can compare lifetimes of two ( n − k + 1 ) -out-of- n systems with dependent components with respect to usual stochastic ordering. We also consider the Archimedean copula with an n -monotone generator obtained by gamma distribution (which generates Gamma-Simplex Copulas described in McNeil and Nešlehovà (2010) [19]). The cumulative distribution function (cdf) of the lifetime of an ( n − k + 1 ) -out-of- n system with dependent components is also obtained. Then, some trivial conditions under which one can compare lifetimes of two ( n − k + 1 ) -out-of- n systems in this case are provided. The cdf of order statistics arising from a random vector whose dependence structure is described by an Archimedean copula with a cm generator is also obtained. Some simple conditions under which one can compare lifetimes of two ( n − k + 1 ) -out-of- n systems in this case are investigated. Finally, we shall generalize the results of Ma (1997), which compare lifetimes of two ( n − k + 1 ) -out-of- n systems with heterogeneous dependent populations and homogeneous dependent populations, for samples with dependent components.